Complex hamiltoniandynamics

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Tassos Bountis � Haris Skokos

Complex HamiltonianDynamics

With a Foreword by Sergej Flach

123

Tassos BountisUniversity of PatrasDepartment of MathematicsPatrasGreece

Haris SkokosMPI for the Physics of Complex SystemsDresdenGermanyandPhysics DepartmentAristotle University of ThessalonikiThessalonikiGreece

ISSN 0172-7389ISBN 978-3-642-27304-9 e-ISBN 978-3-642-27305-6DOI 10.1007/978-3-642-27305-6Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2012935685

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Springer is part of Springer Science+Business Media (www.springer.com)

To our wives Angela and Irini, our childrenAthena and Dimitris and our parents Athenaand Christos, Niki and Dimitris.Their love, dedication and support are aconstant source of inspiration andencouragement in our lives.

Foreword

What makes a science book interesting, valuable, useful, and perhaps also worthspending the money to own it? Well, in the case of this book I guess it starts withits title which immediately attracts one’s attention. Next, the intrigued reader isinvited to inspect a Contents List which combines many familiar-sounding topicsand chapters along with completely new and unknown ones, raising our curiosity tofind out how these different topics are interrelated. Ultimately, of course, we realizethat it is the reading itself which will tell us whether we have found enough newand interesting insights into this domain of the endless world of wonders in science,which the authors call complex Hamiltonian dynamics. This brief foreword is thusdevoted to tell the reader that all the above conditions are satisfied for the book youhold in your hands (or have downloaded to your electronic device).

To begin with, the short title tells those of you who do not know it already thatHamiltonian dynamics is endlessly complex. Indeed, Hamiltonian models can beformally used for almost any problem in nature, which includes hopelessly complexsystems as well. But complexity in Hamiltonian dynamics starts on much lower andseemingly simpler levels. Just get into Chap. 2, where the authors demonstrate in apedagogical and very readable way some basic and well-known facts of chaos theoryfor systems with a few degrees of freedom, illustrated by a number of illuminatingexamples.

While reading the chapters that follow, the concept of the monograph, its title,and the intentions of the authors quickly become clear. In a nutshell, what isaddressed here is the border between strongly chaotic and fully regular dynamicalsystems. The authors use recent progress in the study of relaxation of nonequi-librium states as a playground for applying novel tools. The models are carefullychosen from a set of well-known half-century-old paradigms, which were inventedto address basic questions of statistical physics. The fact that a number of thesequestions still remain unanswered hints at a kind of complexity that is present atseemingly simple levels. It is indeed worth noting that the experienced authors takespecial care to formulate a number of exercises, which makes this monograph acombination of an introduction into the nonlinear dynamics of many degrees of

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freedom, a report on recent progress at the forefront of nonlinear science, and anideal textbook for students and teachers of advanced physics courses.

An interesting attempt to describe and distinguish order from chaos in Hamil-tonian systems with many degrees of freedom is given in Chap. 3. The authorsdiscuss various types of fixed and dynamical equilibria and their local stabilityproperties, and smoothly connect them to the issue of global stability of dynamicalstates. Besides discussing standard ways of characterizing chaos via Lyapunovspectra, the authors also introduce novel methods (called SALI and GALI) whichconnect tangent dynamics with stability of motion and the nature of the dynamicalstate under investigation. Both aspects of characterizing equilibria and exploringtechniques to distinguish order from chaos are discussed in detail in the twosubsequent chapters.

Then comes Chap. 6, where we encounter many applications of the methodsintroduced earlier to the classical problem of the FPU paradox, first posed by EnricoFermi, John Pasta, and Stanislav Ulam in their pioneering studies of the early 1950s.The next chapter addresses the fascinating phenomenon of localization and reportson another recent puzzle of great interest at the border between order and chaos,namely, the spreading of nonlinear waves in ordered and disordered media.

The monograph continues in Chap. 8 with a critical discussion and comparisonof many systems exhibiting “weak chaos” from the viewpoint of nonextensivestatistical mechanics. Finally, the book ends in Chap. 9 with a number of openproblems, which should prove quite inspiring to graduate student readers, andconcludes with a brief review of additional fascinating topics of Hamiltoniandynamics, which are of great current interest and outstanding potential for practicalapplications.

In summary, this book constitutes in my opinion a very solid piece of work,which serves several purposes: It is very useful as an introductory textbook thatfamiliarizes the reader with modern methods of analysis applied to Hamiltoniansystems of many degrees of freedom and reviews a set of modern research areasat the forefront of nonlinear science. Last but not least, thanks to its pedagogicalstructure, it should prove easily exploitable as an exercise source for advanceduniversity courses.

Dresden, Germany Sergej Flach

Preface

The main purpose of this book is to present and discuss, in an introductory andpedagogical way, a number of important recent developments in the dynamicsof Hamiltonian systems of N degrees of freedom. This is a subject with a longand glorious history, which continues to be actively studied due to its manyapplications in a wide variety of scientific fields, the most important of thembeing classical mechanics, astronomy, optics, electromagnetism, solid state physics,quantum mechanics, and statistical mechanics.

One could, of course, immediately point out the absence of biology, chemistry, orengineering from this list. And yet, even in such diverse areas, when the oscillationsof mutually interacting elements arise, a Hamiltonian formulation can proveespecially useful, as long as dissipation phenomena can be considered negligible.This situation occurs, for example, in weakly oscillating mechanical structures, low-resistance electrical circuits, energy transport processes in macromolecular modelsof motor proteins, or vibrating DNA double helical structures.

Let us briefly review some basic facts about Hamiltonian dynamics, beforeproceeding to describe the contents of this book.

The fundamental property of Hamiltonian systems is that they are derived fromHamilton’s Principle of Least Action and are intimately related to the conservationof book, under time evolution in the phase space of their position and momentumvariables qk , pk , k D 1; 2; : : : ; N , defined in the Euclidean phase space R2N . Theirassociated system of (first-order) differential equations of motion is obtained from aHamiltonian function H , which depends on the phase space variables and perhapsalso time. If H is explicitly time-independent, it represents a first integral of themotion expressing the conservation of total energy of the Hamiltonian system. Thedynamics of this system is completely described by the solutions (trajectories ororbits) of Hamilton’s equations, which lie on a .2N � 1/-dimensional manifold, theso-called energy surface, H.q1; : : : ; qN ; p1; : : : ; pN / D E .

This constant energy manifold can be compact or not. If it is not, some orbits mayescape to infinity, thus providing a suitable framework for studying many problemsof interest to the dynamics of scattering phenomena. In the present book, however,we shall be exclusively concerned with the case where the constant energy manifold

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is compact. In this situation, the well-known theorems of Liouville-Arnol’d (LA)and Kolmogorov-Arnol’d-Moser (KAM) rigorously establish the following twoimportant facts [19].

The LA theorem: If N � 1 global, analytic, single-valued integrals exist (besidesthe Hamiltonian) that are functionally independent and in involution (the Poissonbracket of any two of them vanishes), the system is called completely integrable,as its equations can in principle be integrated by quadratures to a single integralequation expressing the solution curves. Moreover, these curves generally lieon N -dimensional tori and are either periodic or quasiperiodic functions of Nincommensurate frequencies.

The KAM theorem: If H can be written in the form H D H0 C "H1 of an" perturbation of a completely integrable Hamiltonian system H0, most (in thesense of positive measure) quasiperiodic tori persist for sufficiently small ". Thisestablishes the fact that many near-integrable Hamiltonian systems (like the solarsystem for example!) are “globally stable” in the sense that most of their solutionsaround an isolated stable-elliptic equilibrium point or periodic orbit are “regular” or“predictable.”

And what about Hamiltonian systems which are far from integrable? As hasbeen rigorously established and numerically amply documented, they possess neartheir unstable equilibria and periodic orbits dense sets of solutions which are calledchaotic, as they are characterized by an extremely sensitive dependence on initialconditions known as chaos. These chaotic solutions also exist in generic near-integrable Hamiltonian systems down to arbitrarily small values of " ! 0 andform a network of regions on the energy surface, whose size generally grows withincreasing j"j.

In the last four decades, since KAM theory and its implications became widelyknown, Hamiltonian systems have been studied exhaustively, especially in thecases of N D 2 and N D 3 degrees of freedom. A wide variety of powerfulanalytical and numerical tools have been developed to (1) verify whether a givenHamiltonian system is integrable; (2) examine whether a specific initial state leadsto a periodic, quasiperiodic, or chaotic orbit; (3) estimate the “size” of regulardomains of predominantly quasiperiodic motion; and (4) analyze mathematicallythe “boundary” of these regular domains, beyond which large-scale chaotic regionsdominate the dynamics and most solutions exhibit in the course of time statisticalproperties that prevail over their deterministic character.

As it often happens, however, physicists are more daring than mathematicians.Impatient with the slow progress of rigorous analysis and inspired by the pioneeringnumerical experiments of Fermi, Pasta, and Ulam (FPU) in the 1950s, a numberof statistical mechanics experts embarked on a wonderful journey in the field ofN � 1 coupled nonlinear oscillator chains and lattices and discovered a goldmine.Much to the surprise of their more traditional colleagues, they discovered a wealthof extremely interesting results and opened up a path that is most vigorously pursuedto this very day. They concentrated especially on one-dimensional FPU lattices (orchains) of N classical particles and sought to uncover their transport properties,especially in the N ! 1 and t ! 1 limits.

Preface xi

They were joined in their efforts by a new generation of mathematical physicistsaiming ultimately to establish the validity of Fourier’s law of heat conduction,unravel the mysteries of localized oscillations, understand energy transport, andexplore the statistical properties of these Hamiltonian systems at far from equilib-rium situations. They often set all parameters equal, but also seriously ponderedthe effect of disorder and its connections with nonlinearity. Although most resultsobtained to date concern (d D 1)-dimensional chains, a number of findings havebeen extended to the case of higher (d > 1)-dimensional lattices.

Throughout these studies, regular motion has been associated with quasiperiodicorbits onN -dimensional tori and chaos has been connected to Lyapunov exponents,the maximal of which is expected to converge to a finite positive value in the longtime limit t ! 1. Recently, however, this “duality” has been challenged by anumber of results regarding longtime Hamiltonian dynamics, which reveal (a) therole of tori with a dimension as low as d D 2; 3; : : : on the 2N � 1 energy surfaceand (b) the significance of regimes of “weak chaos,” near the boundaries of regularregions. These phenomena lead to the emergence of a hierarchy of structures, whichform what we call quasi-stationary states, and give rise to particularly long-livedregular or chaotic phenomena that manifest a deeper level of complexity with far-reaching physical consequences.

It is the purpose of this book to discuss these phenomena within the context ofwhat we call complex Hamiltonian dynamics. In the chapters that follow, we intendto summarize many years of research and discuss a number of recent results withinthe framework of what is already known about N degrees of freedom Hamiltoniansystems. We intend to make the presentation self-contained and introductory enoughto be accessible to a wide range of scientists, young and old, who possess some basicknowledge of mathematical physics.

We do not intend to focus on traditional topics of Hamiltonian dynamics, suchas their symplectic formalism, bifurcation properties, renormalization theory, orchaotic transport in homoclinic tangles, which have already been expertly reviewedin many other textbooks. Rather, we plan to focus on the progress of the lastdecade on one-dimensional Hamiltonian lattices, which has yielded, in our opinion,a multitude of inspiring discoveries and new insights, begging to be investigatedfurther in the years to come.

More specifically, we propose to present in Chap. 1 some fundamental back-ground material on Hamiltonian systems that would help the uninitiated readerbuild some basic knowledge on what the rest of the book is all about. As partof this introductory material, we mention the pioneering results of A. Lyapunovand H. Poincare regarding local and global stability of the solutions of Hamiltoniansystems. We then consider in Chap. 2 some illustrative examples of Hamiltoniansystems ofN D 1 and 2 degrees of freedom and discuss the concept of integrabilityand the departure from it using singularity analysis in complex time and perturbationtheory. In particular, the occurrence of chaos in such systems as a result ofintersections of invariant manifolds of saddle points will be examined in some detail.

In Chap. 3, we present in an elementary way the mathematical concepts and basicingredients of equilibrium points, periodic orbits, and their local stability analysis

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for arbitrary N . We describe the method of Lyapunov exponents and examine theirusefulness in estimating the Kolmogorov entropy of certain physically importantHamiltonian systems in the thermodynamic limit, that is, taking the total energy Eand the number of particles N very large with E=N D constant. Moreover, weintroduce some alternative methods for distinguishing order from chaos based onthe more recently developed approach of Generalized Alignment Indices (GALIs)described in detail in Chap. 5.

Chapter 4 introduces the fundamental notions of nonlinear normal modes(NNMs), resonances, and their implications for global stability of motion inHamiltonian systems with a finite number of degrees of freedom N . In particular,we examine the importance of discrete symmetries and the usefulness of grouptheory in analyzing periodic and quasiperiodic motion in Hamiltonian systems withperiodic boundary conditions. Next, we discuss in Chap. 5 a number of analyticaland numerical results concerning the GALI method (and its ancestor the SmallerAlignment Index—SALI—method), which uses properties of wedge products ofdeviation vectors and exploits the tangent dynamics to provide indicators of stableand chaotic motion that are more accurate and efficient than those proposed byother approaches. All this is then applied in Chap. 6 to explain the paradox ofFPU recurrences and the associated transition from “weak” to “strong” chaos. Weintroduce the notion of energy localization in normal mode space and discuss theexistence and stability of low-dimensional “q-tori,” aiming to provide a more com-plete interpretation of FPU recurrences and their connection to energy equipartitionin FPU models of particle chains.

In Chap. 7 we proceed to discuss the phenomenon of localized oscillationsin the configuration space of nonlinear one-dimensional lattices with N ! 1,concentrating first on the so-called periodic (or translationally invariant) case whereall parameters in the on-site and interaction potentials are identical. We also mentionin this chapter recent results regarding the effects of delocalization and diffusiondue to disorder introduced by choosing some of the parameters (masses or springconstants) randomly at the initialization of the system.

Next, in Chap. 8 we examine the statistical properties of chaotic regions in caseswhere the orbits exhibit “weak chaos,” for example, near the boundaries of islandsof regular motion where the positive Lyapunov exponents are relatively small.We demonstrate that “stickiness” phenomena are particularly important in theseregimes, while probability density functions (pdfs) of sums of orbital components(treated as random variables in the sense of the Central Limit Theorem) are wellapproximated by functions that are far from Gaussian! In fact, these pdfs closelyresemble q-Gaussian distributions resulting from minimizing Tsallis’ q-entropy(subject to certain constraints) rather than the classical Boltzmann Gibbs (BG)entropy and are related to what has been called nonextensive statistical mechanicsof strongly correlated dynamical processes.

In this context, we discuss chaotic orbits close to unstable NNMs of multidimen-sional Hamiltonian systems and show that they give rise to certain very interestingquasi-stationary states, which last for very long times and whose pdfs (of the abovetype) are well fitted by functions of the q-Gaussian type. Of course, in most cases, as

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t continues to grow, these pdfs are expected to converge to a Gaussian distribution(q ! 1), as chaotic orbits exit from weakly chaotic regimes into domains ofstrong chaos, where the positive Lyapunov exponents are large and BG statisticsprevail. Still, we suggest that the complex statistics of these states need to beexplored further, particularly with regard to the onset of energy equipartition, astheir occurrence is far from exceptional and their long-lived nature implies that theymay be physically important in unveiling some of the mysteries of Hamiltoniansystems in many dimensions.

The book ends with Chap. 9 containing our conclusions, a list of open researchproblems, and a discussion of future prospects in a number of areas of Hamiltoniandynamics. Moreover, at the end of every chapter we have included a number ofexercises and problems aimed at training the uninitiated reader to learn how to usesome of the fundamental concepts and techniques described in this book. Some ofthe problems are intended as projects for ambitious postgraduate students and offersuggestions that may lead to new discoveries in the field of complex Hamiltoniandynamics in the years ahead.

In the Acknowledgments that follow this Preface, we express our gratitude to anumber of junior and senior scientists, who have contributed to the present book inmany ways: Some have provided useful comments and suggestions on many topicstreated in the book, while others have actively collaborated with us in obtainingmany of the results presented here.

Whether we have done justice to all those whose work is mentioned in the textand listed in our References is not for us to judge. The fact remains that, beyond thehelp we have received from all acknowledged scientists and referenced sources, theresponsibility for the accurate presentation and discussion of the scientific field ofcomplex Hamiltonian dynamics lies entirely with the authors.

Patras, Greece Tassos BountisDresden, Germany Haris Skokos

Acknowledgments

Throughout our career and, in particular, in the course of our work described in thepresent book, we have been privileged to collaborate with many scientists, juniorand senior to us in age and scientific experience. It is impossible to do justiceto all of them in the limited space available here. Among the senior scientists,Tassos Bountis (T. B.) first wishes to mention Professors Elliott Montroll and RobertHelleman of the University of Rochester, New York, and Professor Joseph Ford ofGeorgia Institute of Technology, who first introduced him to the wonderful worldof Hamiltonian Systems and Nonlinear Dynamics in the USA. Most influentialto his particular scientific viewpoint have also been Professors Gregoire Nicolis,Universite Libre de Bruxelles, Giulio Casati, University of Insubria, and HansCapel, University of Amsterdam, and, in Greece, Professors George Contopoulos,University of Athens, and John Nicolis, University of Patras. More recently, T. B.has worked on lattice dynamics based on a number of ideas and results put forth byDr. Sergej Flach and his group at the Max Planck Institute of the Physics of ComplexSystems in Dresden, Professor George Chechin of the Southern Federal Universityof Russia at Rostov-on-Don, and Professor Constantino Tsallis and his coworkersat the Centro Brasileiro de Pesquisas Fisicas in Rio de Janeiro.

T. B. has benefited significantly from the collaboration of several of his Ph.D.students, whose ingenuity, enthusiasm, and hard work produced many of the resultsdiscussed in this book. Their dynamic personalities, congeniality, and pleasantcharacter created a most enjoyable and productive environment at the Department ofMathematics and the Center of Research and Applications of Nonlinear Systems ofthe University of Patras. T. B., therefore, happily acknowledges many enlighteningdiscussions with Vassilis Rothos, Jeroen Bergamin, Christos Antonopoulos, HelenChristodoulidi, and Thanos Manos, whose work has contributed significantly to thecontents of this book. We want to thank them all especially for allowing us to usemany figures and results of our common publications in this book.

Haris Skokos (H. S.) also wishes to express his gratitude to a number of scientistswho have played a significant role in helping him shape his career as a researcher.More specifically, following a chronological order of his collaborations, he wishes to

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mention the late Dr. Chronis Polymilis of the University of Athens, who introducedhim to the fascinating world of Nonlinear Dynamics during his undergraduate years.Later on, he had the opportunity to closely work with Professor Antonio Giorgilli ofthe University of Milan, Dr. Panos Patsis of the Academy of Athens, and Dr. LiaAthanassoula of the Observatory of Marseilles, on applications of Hamiltoniandynamics to problems of astronomical interest. H. S. wishes to express his deepestthanks to all of them, both for their friendship and productive collaboration whichhelped him deepen his knowledge in several theoretical aspects of Chaos Theoryand enhanced his ability to work on applications to real physical problems.

The next stop of H. S.’s journey in the tangled world of Chaotic Dynamics wasParis, France, where he had the privilege to work together with Professor JacquesLaskar at the Observatory of Paris. H. S. is grateful to Jacques for broadening hisknowledge on chaos detection methods (which is one of his most loved subjects)and symplectic integration techniques as well as for giving him the opportunity towork on the dynamics of real accelerators.

H. S. also considers himself most fortunate for having been able to work inthe multidisciplinary environment of the Max Planck Institute of the Physics ofComplex Systems in Dresden, where this book was written. He expresses hisgratitude to Dr. Sergej Flach not only for giving him the opportunity to extend hisactivity to the intriguing field of disordered nonlinear systems, but also for his truefriendship which made H.S.’s stay in Dresden one of the most pleasant periods ofhis life.

We also wish to thank Professor Ko van der Weele for critically reading themanuscript and providing many suggestions, Dr. Enrico Gerlach for preparingthe MAPLE algorithms for the computation of the SALI and GALI presentedin Chap. 5, and Dr. Yannis Kominis for helping us summarize applications ofHamiltonian dynamics to nonlinear optics and plasma physics, in the last part ofChap. 9.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preamble .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Lyapunov Stability of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Hamiltonian Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Complex Hamiltonian Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Hamiltonian Systems of Few Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . 132.1 The Case of N D 1 Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The Case of N D 2 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Coordinate Transformations and Solution by Quadratures . . . 202.2.2 Integrability and Solvability of the Equations of Motion . . . . . 24

2.3 Non-autonomous One Degree of Freedom Hamiltonian Systems . . . . 302.3.1 The Duffing Oscillator with Quadratic Nonlinearity .. . . . . . . . . 312.3.2 The Duffing Oscillator with Cubic Nonlinearity . . . . . . . . . . . . . . 34

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Local and Global Stability of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 Equilibrium Points, Periodic Orbits and Local Stability . . . . . . . . . . . . . . 41

3.1.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.2 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.1 An Analytical Criterion for “Weak” Chaos. . . . . . . . . . . . . . . . . . . . 51

3.3 Lyapunov Characteristic Exponents and “Strong” Chaos . . . . . . . . . . . . . 533.3.1 Lyapunov Spectra and Their Convergence . . . . . . . . . . . . . . . . . . . . 53

3.3.1.1 Lyapunov Spectra and theThermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Distinguishing Order from Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 The SALI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.2 The GALI Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


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4 Normal Modes, Symmetries and Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Normal Modes of Linear One-Dimensional Hamiltonian Lattices . . . 634.2 Nonlinear Normal Modes (NNMs) and the Problem

of Continuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Periodic Boundary Conditions and Discrete Symmetries . . . . . . . . . . . . . 67

4.3.1 NNMs as One-Dimensional Bushes . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.2 Higher-Dimensional Bushes and Quasiperiodic Orbits . . . . . . . 70

4.4 A Group Theoretical Study of Bushes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4.1 Subgroups of the Parent Group and Bushes

of NNMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.4.2 Bushes in Modal Space and Stability Analysis. . . . . . . . . . . . . . . . 74

4.5 Applications to Solid State Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5.1 Bushes of NNMs for a Square Molecule . . . . . . . . . . . . . . . . . . . . . . 804.5.2 Bushes of NNMs for a Simple Octahedral Molecule . . . . . . . . . 84


5 Efficient Indicators of Ordered and Chaotic Motion . . . . . . . . . . . . . . . . . . . . 915.1 Variational Equations and Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 The SALI Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 The GALI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.1 Theoretical Results for the Time Evolution of GALI . . . . . . . . . 1045.3.1.1 Exponential Decay of GALI for Chaotic Orbits . . . . 1045.3.1.2 The Evaluation of GALI for Ordered Orbits . . . . . . . . 107

5.3.2 Numerical Verification and Applications . . . . . . . . . . . . . . . . . . . . . . 1115.3.2.1 Low-Dimensional Hamiltonian Systems . . . . . . . . . . . . 1115.3.2.2 High-Dimensional Hamiltonian Systems . . . . . . . . . . . 1175.3.2.3 Symplectic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3.2.4 Motion on Low-Dimensional Tori . . . . . . . . . . . . . . . . . . . 122

Appendix A: Wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Appendix B: Example algorithms for the computation of the

SALI and GALI chaos indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 FPU Recurrences and the Transition from Weak to Strong Chaos . . . . 1336.1 The Fermi Pasta Ulam Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.1.2 The Concept of q-breathers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.1.3 The Concept of q-tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Existence and Stability of q-tori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.2.1 Construction of q-tori by Poincare-Linstedt Series . . . . . . . . . . . 1406.2.2 Profile of the Energy Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3 A Numerical Study of FPU Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.3.1 Long Time Stability near q-tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Contents xix

6.4 Diffusion and the Breakdown of FPU Recurrences . . . . . . . . . . . . . . . . . . . 1576.4.1 Two Stages of the Diffusion Process. . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.4.2 Rate of Diffusion of Energy to the Tail Modes . . . . . . . . . . . . . . . . 1606.4.3 Time Interval for Energy Equipartition .. . . . . . . . . . . . . . . . . . . . . . . 162


7 Localization and Diffusion in Nonlinear One-Dimensional Lattices . . . 1657.1 Introduction and Historical Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.1.1 Localization in Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.2 Discrete Breathers and Homoclinic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 168

7.2.1 How to Construct Homoclinic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.3 A Method for Constructing Discrete Breathers . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3.1 Stabilizing Discrete Breathers by a Control Method .. . . . . . . . . 1757.4 Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.4.1 Anderson Localization in Disordered Linear Media . . . . . . . . . . 1807.4.2 Diffusion in Disordered Nonlinear Chains . . . . . . . . . . . . . . . . . . . . 183

7.4.2.1 Two Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.4.2.2 Regimes of Wave Packet Spreading . . . . . . . . . . . . . . . . . 1857.4.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186


8 The Statistical Mechanics of Quasi-stationary States . . . . . . . . . . . . . . . . . . . . 1918.1 From Deterministic Dynamics to Statistical Mechanics . . . . . . . . . . . . . . 191

8.1.1 Nonextensive Statistical Mechanics andq-Gaussian pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2 Statistical Distributions of Chaotic QSS and Their Computation . . . . 1958.3 FPU �-Mode Under Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . 197

8.3.1 Chaotic Breathers and the FPU �-Mode . . . . . . . . . . . . . . . . . . . . . . 2008.4 FPU SPO1 and SPO2 Modes Under Fixed Boundary Conditions .. . . 2038.5 q-Gaussian Distributions for a Small Microplasma System . . . . . . . . . . 2088.6 Chaotic Quasi-stationary States in Area-Preserving Maps . . . . . . . . . . . . 212

8.6.1 Time-Evolving Statistics of pdfs in Area-Preserving Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.6.2 The Perturbed MacMillan Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148.6.2.1 The " D 0:9, � D 1:6 Class of Examples . . . . . . . . . . . 2158.6.2.2 The � D 1:2, � D 1:6 Class of Examples . . . . . . . . . . . 217


9 Conclusions, Open Problems and Future Outlook . . . . . . . . . . . . . . . . . . . . . . . 2219.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.2.1 Singularity Analysis: Where Mathematics Meets Physics . . . . 224

xx Contents

9.2.2 Nonlinear Normal Modes, Quasiperiodicityand Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

9.2.3 Diffusion, Quasi-stationary States and Complex Statistics . . . 2299.3 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.3.1 Anomalous Heat Conduction and Control of Heat Flow . . . . . 2319.3.2 Complex Soliton Dynamics in Nonlinear Photonic

Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.3.3 Kinetic Theory of Hamiltonian Systems and

Applications to Plasma Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Acronyms

BEC Bose Einstein Condensation: Important physical phenomenon related tothe behavior of bosons at very low temperatures.

BG Boltzmann Gibbs: Ensembles of classical equilibrium statisticalmechanics proposed by Boltzmann and Gibbs in the late nineteenthand early twentieth centuries.

CLT Central Limit Theorem: The classical theorem, according to which sumsofN identically and independently distributed random variables tend, inthe limit N ! 1, to a Gaussian distribution with the same mean andvariance as the original variables.

CPU Central Processing Unit: The part of a computer system that performsthe basic arithmetical, logical, and input/output operations required bya computer program. The time required by the CPU for the completionof the tasks of a particular computer program (CPU time) is used tocharacterize the efficiency of the program.

DNLS Discrete Nonlinear Schrodinger Equation: The Hamiltonian system ofequations emanating from the NLS by the discretization of its secondderivative with respect to the space variable.

DDNLS Disordered Discrete Nonlinear Schrodinger Equation: The DNLS whensome of its parameters are chosen randomly and uniformly from withina specified interval.

dof Degree(s) of freedom: The number of canonical conjugate pairs ofposition and momentum variables characterizing a Hamiltonian system.

FPU Fermi, Pasta and Ulam: Names of the researchers who first integratednumerically a chain (one-dimensional lattice) of identical oscillatorscoupled by quadratic as well as cubic and/or quartic nearest neighborinteraction terms in their Hamiltonian.

FPU�˛ FPU lattice whose interaction terms in the Hamiltonian, beyond theharmonic ones, are only of the cubic type.

FPU�ˇ FPU lattice whose interaction terms in the Hamiltonian, beyond theharmonic ones, are only of the quartic type.

xxi

xxii Acronyms

GALIk Generalized Alignment Index of order k � 2: Chaos indicator relatedto the book of the parallelepiped formed by k deviation vectors in thetangent space of an orbit of a dynamical system.

IPM In Phase Mode: A particular NNM of one-dimensional lattices, whereall particles oscillate identically and in phase.

KAM Kolmogorov Arnol’d Moser: Name of a theorem that establishes animportant rigorous result regarding the behavior of weakly perturbedintegrable Hamiltonian systems.

KdV Korteweg de Vries: Name of an equation exhibiting solitary wavesolutions first obtained by Korteweg and de Vries in the 1890s. In the1960s these waves were named solitons and formed an integral part ofthe theory of completely integrable evolution equations.

KG Klein Gordon: It refers to the so-called Klein Gordon potential ofclassical and quantum physics, which consists of a quadratic and aquartic part.

LA Liouville Arnol’d: Name of a theorem that establishes an importantrigorous result regarding the solvability of integrable Hamiltonian sys-tems of N dof and the existence of N -dimensional tori on which theirbounded solutions lie.

LCE(s) Lyapunov Characteristic Exponent(s): Exponents characterizing therates of divergence of nearby trajectories of dynamical systems in phasespace.

MLE Maximum Lyapunov Exponent: The LCE with the maximal value.NLS Nonlinear Schrodinger Equation: A completely integrable PDE which

consists of the linear Schrodinger equation plus a cubic nonlinearity andis of particular importance in the field of nonlinear optics.

NNM(s) Nonlinear Normal Mode(s): Extension of the normal modes of a linearsystem of coupled harmonic oscillators in the nonlinear regime.

NME(s) Normal Mode Eigenvector(s): Eigenvector(s) of a linear problem, usedas a basis for the expansion of the solutions of the associated perturbednonlinear problem.

ODE(s) Ordinary Differential Equation(s): Differential equations involvingderivatives with respect to only one independent variable.

OPM Out of Phase Mode: A particular NNM of one-dimensional lattices,where all particles oscillate identically out of phase with respect to eachother in all neighboring pairs.

PDE(s) Partial Differential Equation(s): Differential equations involving deriva-tives with respect to more than one independent variable.

pdf(s) Probability distribution function(s): Function(s) representing the prob-ability density of observables related to the long time evolution ofvariables of a dynamical system in chaotic regimes.

PSS Poincare Surface of Section: Cross-section of orbits of a Hamiltoniansystem with certain lower dimensional subspace of its phase space. Forexample, in the case of a two dof Hamiltonian system the PSS is atwo-dimensional plane defined by one pair of position and momentum

Acronyms xxiii

coordinates, say .x; px/, at times when the second position coordinatehas a prescribed value, say y D 0, and the second momentum aprescribed sign, say py > 0.

QSS Quasi-stationary states: Weakly chaotic dynamical states of Hamiltoniansystems that persist for very long times and are generally characterizedby pdfs that are different from Gaussian.

SALI Smaller Alignment Index: Chaos indicator related to the area of aparallelogram formed by two deviation vectors in the tangent space ofan orbit of a dynamical system. It is analogous to GALI2.

SPO(s) Simple Periodic Orbit(s): Periodic orbit(s) of a dynamical system return-ing to their initial state after a single oscillation of its variables.

Chapter 1Introduction

Abstract Chapter 1 starts by defining a dynamical system in terms of ordinarydifferential equations and presents the fundamental framework within which one canstudy the stability of their equilibrium (or fixed) points, as developed by the greatRussian mathematician A. M. Lyapunov. The concept of Lyapunov CharacteristicExponents is introduced and two theorems by Lyapunov are discussed, whichestablish criteria for the asymptotic stability of a fixed point. The mathematicalsetting of a Hamiltonian system is presented and a third theorem by Lyapunovis stated concerning the continuation of the linear normal modes of N harmonicoscillators, when the system is perturbed by adding to the Hamiltonian nonlinearterms higher than quadratic. Finally, we discuss the meaning of complexity inHamiltonian dynamics, by referring to certain weakly chaotic orbits, which formcomplicated quasi-stationary states that are well-approximated by the principles ofnonextensive statistical mechanics for very long times.

1.1 Preamble

The title of this book consists of three words and it is important to understand allof them. We shall not start, however, with the first one, as it is the most difficult todefine. It may also turn out to be the most fascinating and intriguing one in the end.Rather, we will start with the last word: What is dynamics?

Well, it comes from the Greek word “dynamis” meaning “force”, so it will bevery natural to think of it in the Newtonian context of mechanics as describingthe effect of a vectorial quantity proportional to the mass and acceleration of agiven body. But that is a very limited interpretation. Dynamics, in fact, refers to anyphysical observable that is in motion with respect to a stationary frame of reference.And as every student knows this motion may very well be uniform (i.e. with constantvelocity) and hence not involve any force at all.

Thus we will speak of dynamics when we are interested to know how the stateof an observable changes in time. This may represent, for example, the position or

T. Bountis and H. Skokos, Complex Hamiltonian Dynamics,Springer Series in Synergetics, DOI 10.1007/978-3-642-27305-6 1,© Springer-Verlag Berlin Heidelberg 2012

1

2 1 Introduction

velocity of a mass particle in the three-dimensional space, the flow of current in anelectrical circuit, the concentration of a chemical substance, or the population ofa biological species. When we refer to a collection of individual components, weshall speak of the dynamics of a system. Most frequently, these components willinteract with each other and act interdependently. And this is when matters start toget complicated.

What is the nature of this interdependence and how does it affect the dynamics?Does it always lead to behavior that we call “unpredictable”, or chaotic (in thesense of extremely sensitive dependence on initial conditions), or does it also giverise to motions that we may refer to as “regular”, “ordered” or “predictable”? Andwhat about the intermediate regime between order and chaos, where complexityfrequently lies?

It is clear that if we wish to address such questions, we must first introduce aproper mathematical framework for their study and this is provided by the fieldof Ordinary Differential Equations (ODEs). In this context, we will assume thatour system possesses n real dependent variables .xk.t/; k D 1; 2; : : : ; n, whichconstitute a state x.t/ D .x1.t/; : : : ; xn.t// in the phase space of the system D �R

n and are functions of the single independent variable of the problem: the timet 2 R. Their dynamics is described by the system of first order ODEs

dxk

dtD fk.x1; x2; : : : ; xn/; k D 1; 2; : : : ; n; (1.1)

which is thus called a dynamical system. Since the fk do not explicitly depend on t

the system is called autonomous. The functions fk are defined everywhere in D and,for simplicity, we shall assume that they are analytic in all their variables, meaningthat they can be expressed as convergent series expansions in the xk (with non-zeroradius of convergence) near one of their equilibrium (or fixed) points, located at theorigin of phase space 0 D .0; : : : ; 0/ 2 D, where

fk.0/ D 0; k D 1; 2; : : : ; n: (1.2)

This implies that the right sides of our (1.1) can be expanded in power series:

dxk

dtD pk1x1 C pk2x2 C : : : C pknxn C

X

m1;m2;:::;mn

P.m1;:::;mn/

k xm1

1 xm2

2 : : : xmnn (1.3)

where k D 1; 2; : : : ; n and the mk > 0 are such that m1 C m2 C : : : C mn > 1.Since the fk are analytic, the existence and uniqueness of the solutions of (1.1), forany initial condition x1.0/; : : : ; xn.0/ (where the fk are defined) is guaranteed bythe classical theorems of Euler, Cauchy and Picard [46,97]. Furthermore, the seriesin (1.3) are convergent for jxkj � Ak and by a well-known theorem of analysis(e.g. see [155, p. 273]) its coefficients are bounded by:

j P.m1;:::;mn/

k j� Mk

A1m1A2

m2 : : : Anmn

; (1.4)

1.2 Lyapunov Stability of Dynamical Systems 3

where Mk is an upper bound of the modulus of all terms of the form xm1

1 xm2

2 : : : xmnn

entering in (1.3).What can we say about the solutions of these equations in a small neighborhood

of the equilibrium point (1.2), where the series expansions of the fk converge? Is themotion “regular” or “predictable” there? This is the question of stability of motionfirst studied systematically by the great Russian mathematician A. M. Lyapunov,more than 110 years ago [232]. In the section that follows we shall review his famousmethod that led to the proof of the existence of periodic solutions, as continuationsof the corresponding oscillations of the linearized system of equations.

This method is local in character, as it describes solutions in a very small regionaround the origin and for finite intervals in time. It generalizes the treatment andresults found in Poincare’s thesis [273], of which Lyapunov learned at a later time,as he explains in the Introduction of [232].

1.2 Lyapunov Stability of Dynamical Systems

Let us first define the notions of stability that Lyapunov had in mind: The first andsimplest one concerns what may be called asymptotic stability, as it refers to thecase where all solutions xk.t/ of (1.3), starting within a domain of the origin givenby jxk.t0/j � Ak , tend to 0 as t ! 1. A less restrictive situation arises when wecan prove that for every 0 < " < "0, no matter how small, all solutions starting att D t0 within a neighborhood of the origin K."/ j B."/, where B."/ is a “ball” ofradius " around the origin, remain inside B."/ for all t � t0.

This weaker condition is often called neutral or conditional stability and willbe of great importance to us, as it frequently occurs in conservative dynamicalsystems (among which are the Hamiltonian ones). These conserve phase spacevolume and hence cannot come to a complete rest at any value of t , finite or infinite(except for some special solutions lying on stable invariant manifolds, as we discusslater). Conditional stability, in fact, characterizes precisely the systems for whichLyapunov could prove the existence of families of periodic solutions around theorigin by relating them directly to a conserved quantity called integral of the motion.For Lyapunov, the existence of integrals was a means to an end, unlike Poincare,who considered integrability as a primary goal in trying to show the global stabilityof the motion of a dynamical system.

Let us illustrate Lyapunov’s method for establishing the stability of a fixed point,by considering the simple example of a one-dimensional system, n D 1, describedby a single ODE of the Riccati type:

dx

dtD �x C x2: (1.5)

The main idea is to write its solution as a series:

x.t/ D x.1/ C x.2/ C : : : C x.n/ C : : : ; (1.6)

4 1 Introduction

whose leading term x.1/ is the general solution of the linear part of (1.5), considerednaturally as the most important part of the solution in a small region around the fixedpoint x D 0. This term is x.1/ D ae�t and incorporates the only arbitrary constantneeded for the general solution of the problem. Substituting (1.6) into (1.5) we thusobtain an infinite set of linear inhomogeneous equations for the x.k/

dx.k/

dtD �x.k/ C

kX

j D1

x.j /x.j �k/; k > 1; (1.7)

from which we can easily obtain particular solutions, since the homogeneous part isalready represented by x.1/.t/. These are: x.k/ D �.�a/ke�kt and hence the generalsolution of (1) becomes:

x.t/ D x.1/ Cx.2/ C : : : D ae�t �a2e�2t Ca3e�3t � : : : D q �q2 Cq3 � : : : ; (1.8)

where we have set q D ae�t . This expression clearly converges for jqj < 1, whichgives the region of initial conditions, at t D 0 with jaj < 1, where these solutionsexist. Furthermore, in this region, all solutions tend to 0 as t ! 1 and therefore 0is an asymptotically stable equilibrium state.

Let us now see how all this works for arbitrary n: First, we write again the generalsolution in the form of a series

xk.t/ D xk.1/ C xk

.2/ C : : : ; k D 1; 2; : : : ; n; (1.9)

and substitute in (1.3) separating the linear system of equations for the xk.1/:

dxk.1/

dtD pk1x

.1/1 C pk2x

.1/2 C : : : C pknx.1/

n ; (1.10)

from the remaining ones

dxk.m/

dtD pk1x

.m/1 C pk2x

.m/2 C : : : C pknx.m/

n C R.m/

k ; m > 1; (1.11)

where the R.m/

k contain only terms x.j /s , s D 1; 2; : : : ; n and j D 1; 2; : : : ; m � 1,

which have already been determined at lower orders. Thus, we first need to studythe linear system (1.10) and obtain its general solution as a linear combination of nindependent particular solutions

x.1/

k .t/ D a1xk1.t/ C a2xk2.t/ C : : : C anxkn.t/; k D 1; 2; : : : ; n; (1.12)

in such a way that the initial conditions xk.1/.t0/ D ak are satisfied. We then use

(1.12) to insert in (1.11) and find particular solutions of the corresponding linearinhomogeneous system for every m, so that xk

.m/.t0/ D 0.

1.2 Lyapunov Stability of Dynamical Systems 5

Lyapunov then obtains explicit expressions for the xk.m/.t/ and makes the

additional assumption that the sets A1; A2; : : : ; An and M1; M2; : : : ; Mn (see thediscussion at the beginning of the chapter), considered as functions of t , have non-zero upper bound for each Ak and non-zero lower bound for each Mk, for allt0 � t � T and any T . Thus, he was able to show that the solution (1.9) written aspower series in the quantities ak is absolutely convergent, as long as the jakj do notexceed some limit depending on T .

Observe now that, even though we have found the general solution, it is impos-sible to discuss the question of stability of the origin, unless we know somethingabout the behavior of the solutions of the linear system (1.10) as functions of t ,as we did for the simple Riccati equation. To proceed one would have to comparefirst every one of the n independent solutions of (1.10) to an exponential functionof time, with the purpose of identifying the particular exponent that would enter insuch a relationship.

To achieve this, Lyapunov had the ingenious idea of introducing what he calledthe characteristic number of a function x.t/, as follows: First form the auxiliaryfunction z.t/ D x.t/e�t and define as the characteristic number �0 of x(t), that valueof � for which z(t) vanishes for � < �0 and becomes unbounded for � > �0, ast ! 1. Thus, this number represents the “rate” of exponential decay (or growth) ofx.t/, as time becomes arbitrarily large. In fact, it is closely connected to the negativeof what we call today the Lyapunov characteristic exponent (LCE) [310]. Adoptingthe convention that functions x.t/ whose z.t/ vanishes for all � have �0 D 1 andthose for which z.t/ is unbounded for all � have �0 D �1, one can thus define aunique characteristic number for every function x.t/.

Proceeding then to derive characteristic numbers for sums and products offunctions arising in the solutions of systems of linear ODEs, Lyapunov defineswhat he calls a regular system of linear ODEs by the condition that the sum ofthe characteristic numbers of its solutions equals the negative of the characteristicnumber of the function

exp

�Z nX

kD1

pkk.t/dt

!(1.13)

(see [232, p. 43]). Thus, he was led to one of his most important theorems:

Theorem 1.1. (see [232, p. 57]) If the linear system of differential equations(1.10) is regular and the characteristic numbers of its independent solutions areall positive, the equilibrium point at the origin is asymptotically stable.

It is important to note that the condition of regularity of a linear system,as defined above, can be shown to hold for all systems, whose coefficientspjk , j; k D 1; 2; : : : ; n are constant or periodic functions of t , and is, in fact,also verified for a much larger class of linear systems. Moreover, since wewill be exclusively concerned here with the case of constant coefficients, it isnecessary to identify the meaning of characteristic numbers for our problem athand. Indeed, as the reader can easily verify, they are directly related to the

6 1 Introduction

eigenvalues of the n � n matrix J D .pjk/; j; k D 1; 2; : : : ; n, obtained as theroots of the characteristic equation

det.J � �In/ D 0; (1.14)

�1; �2; : : : ; �n, In being the n�n identity matrix. In modern terminology, therefore,for a dynamical system (1.1), with an equilibrium point at .0; 0; : : : ; 0/ and aconstant Jacobian matrix

Jk;j D pkj D @fk

@xj

.0; : : : ; 0/; j; k D 1; 2; : : : ; n; (1.15)

the above theorem translates to the following well-known result:

Theorem 1.2. (see [170, p. 181]) If all eigenvalues of the matrix J have negativereal part less than �c; c > 0, there is a compact neighborhood U of the origin,such that, for all .x1.0/; x2.0/; : : : ; xn.0// 2 U , all solutions xk.t/ ! 0, ast ! 1. Furthermore, one can show that this approach to the fixed point isexponential: Indeed, if we denote by j � j the Euclidean norm in R

n and define x.t/ D.x1.t/; x2.t/; : : : ; xn.t//, it can be proved, using simple topological arguments, thatfor all x.0/ 2 U , jx.t/j � jx.0/je�ct , and jx.t/j is in U for all t � 0.

Let us finally observe that, since Lyapunov expressed his solutions as sums ofexponentials of the form

xs.m/ D

XCs

.m1;m2;:::;mn/e.m1�1Cm2�2C:::Cmn�n/t ; (1.16)

with 0 < m1 C m2 C : : : C mn < m, when we substitute these expressions in theequations of motion (1.10), (1.11) and try to solve for the coefficients, Cs

.m1;m2;:::;mn/,we need to divide by the expression

m1�1 C m2�2 C : : : C mn�n: (1.17)

If this quantity ever becomes zero, this will lead to secular terms in the aboveseries which will grow linearly in time. This can happen for example if one (ormore) of the �j D 0, or if the system possesses one (or more) pairs of imaginaryeigenvalues. Note that such so-called resonances will never occur, as long as theconditions of Theorem 1.2 above are satisfied and thus, in the case of Re.�j / > 0

all secular terms are avoided and the convergence of the corresponding series isproved.

Assuming that the linear terms of these series have constant coefficients,Lyapunov then paid particular attention to the case where one (or more) of theeigenvalues of the matrix of these coefficients have zero real part. This wasthe beginning of what we now call bifurcation theory (see e.g. [165, 170]), asit constitutes the turning point between stability of the fixed point (all eigenvalueshave negative real part) and instability (at least one eigenvalue has positive real part).

1.3 Hamiltonian Dynamical Systems 7

Besides the above so-called first method, which is local and parallels similarwork by Poincare, Lyapunov also introduced a second method, which is entirely hisown and is based on the following idea [232]: Instead of focusing on the solutionsof the direct problem, one constructs specific functions of these solutions, withwell-defined geometric properties, whose evolution in time reveals indirectly theproperties of the actual solutions for all t > t0! This is the famous method ofLyapunov functions, as we know it today. It is global in the sense that it does not referto a finite time interval and applies to relatively large regions of phase space aroundthe equilibrium point. However, it does rely strongly on one’s resourcefulnessto construct the appropriate partial differential equation (PDE) satisfied by thesefunctions and solve them by power series expansions.

We will not dwell at all here on Lyapunov’s second method, as it applies almostexclusively to non-conservative (hence non-Hamiltonian) dynamical systems. Theinterested reader can find excellent accounts of the theory and applications ofLyapunov’s functions, not only in Lyapunov’s treatise [232] but also in a numberof excellent textbooks on the qualitative theory of ODEs and dynamical systems[170, 265, 346]. There is, however, one more result of Lyapunov’s theory [232]concerning simple periodic solutions of Hamiltonian systems, which will be veryuseful to us in the chapters that follow and is described separately in the next section.

1.3 Hamiltonian Dynamical Systems

In this book we shall often speak of conditional (or neutral) stability where thereis (at least) one pair of purely imaginary eigenvalues (all others being real andnegative), implying the possibility of the existence of periodic orbits about theorigin. This allows us to apply the above theory to the case of Hamiltoniandynamical systems of N degrees of freedom (dof), where n D 2N and the equationsof motion (1.1) are written in the form

dqk

dtD @H

@pk

;dpk

dtD � @H

@qk

; k D 1; 2; : : : ; N; (1.18)

where qk.t/, pk.t/, k D 1; 2; : : : ; 2N are the position and momentum coordinatesrespectively and H is called the Hamiltonian function. If H does not explicitlydepend on t , it is easy to see from (1.18) that its total time derivative is zero andthus represents a first integral (or constant of the motion), whose value equals thetotal energy of the system E .

In most cases treated in this book, we will assume that our Hamiltonian canbe expanded in power series as a sum of homogeneous polynomials Hm of degreem � 2

H D H2.q1; : : : ; qN ; p1; : : : ; pN / C H3.q1; : : : ; qN ; p1; : : : ; pN / C : : : D E;

(1.19)

8 1 Introduction

so that the origin qk D pk D 0; k D 1; 2; : : : ; N is an equilibrium point of thesystem. H.qk.t/; pk.t// D E thus defines the so-called (constant) energy surface,� .E/ � R

2N , on which our dynamics evolves.Let us now assume that the linear equations resulting from (1.18) and (1.19), with

Hm D 0 for all m > 2, yield a matrix, whose eigenvalues all occur in conjugateimaginary pairs, ˙i!k and thus provide the frequencies of the so-called normalmode oscillations of the linearized system. This means that we can change to normalmode coordinates and write our Hamiltonian in the form of N uncoupled harmonicoscillators

H .2/ D !1

2.x1

2 C y12/ C !2

2.x2

2 C y22/ C : : : C !N

2.x2

N C y2N / D E; (1.20)

where xk; yk; k D 1; 2; : : : ; N are the new position and momentum coordinatesand !k represent the normal mode frequencies of the system.

Theorem 1.3. [232] If none of the ratios of these eigenvalues, !j =!k , is an integer,for any j; k D 1; 2; : : : ; N; j ¤ k, the linear normal modes continue to existas periodic solutions of the nonlinear system (1.18) when higher order termsH3; H4; : : : etc. are taken into account in (1.19).

These solutions have frequencies close to those of the linear modes and areexamples of what we call simple periodic orbits (SPOs), where all variables oscillatewith the same frequency !k D 2�=Tk, returning to the same values after a singlemaximum (and minimum) in their time evolution over one period Tk.

What is the importance of these particular SPOs which are called nonlinearnormal modes, or NNMs? Once we have established that they exist, what can wesay about their stability under small perturbations of their initial conditions? Howdo their stability properties change as we vary the total energy E in (1.19)? Do suchchanges only affect the motion in the immediate vicinity of the NNMs or can theyalso influence the dynamics of the system more globally? Are there other simpleperiodic orbits of comparable importance that may also be useful to study from thispoint of view? These are the questions we shall try to answer in the chapters thatfollow.

After discussing in Chap. 2 the fundamental concepts of order and chaos inHamiltonian systems of few dof, we shall examine in Chap. 3 the topics of localand global stability in the case of arbitrary (but finite) number of dof N . It is herethat we will introduce the spectrum of LCEs for Hamiltonian systems, followingthe discussion of the previous section. In particular, we shall dwell on the relationof the LCEs to the chaotic behavior of individual solutions or orbits of the systemand use them to evaluate the so-called Kolmogorov entropy. More specifically, wewill demonstrate on concrete examples that this entropy is extensive in regimes of“strong chaos”, i.e. it grows linearly with N in the thermodynamic limit (E ! 1,N ! 1 with E=N fixed) and is hence compatible with Boltzmann Gibbs (BG)statistical mechanics.

Then in Chap. 4 we focus on the importance of NNMs in Hamiltonian systemspossessing discrete symmetries and discuss their relevance with regard to physicalmodels oscillating in one, two or three spatial dimensions. Our main example will

1.3 Hamiltonian Dynamical Systems 9

be the Fermi Pasta Ulam (FPU) chain under periodic boundary conditions, but theresults will be quite general and can apply to a wide variety of systems, which areof interest to solid state physics. Next, we will describe in detail in Chap. 5 themethod of the Generalized Alignment Indices GALIk, k D 1; 2; : : : ; 2N indicators.The GALIs are ideal for identifying domains of chaos and order not only in N

dof Hamiltonian systems but also 2N -dimensional (2ND) symplectic maps andrepresent a significant improvement over many other similar techniques used in theliterature. For example, these indicators can also be employed to locate s.� N /-dimensional tori on which the motion is governed by s rationally independentfrequencies and provide conditions for the “breakdown” of these tori and hencethe destabilization of the corresponding quasiperiodic orbits.

In Chap. 6, we return to a very important physical property of Hamiltonianlattices, namely arrays of nonlinear oscillators coupled to each other by nearestneighbor interactions in one or more spatial dimensions. In particular, we revisitthe famous FPU lattice and investigate the phenomenon of FPU recurrences atlow energies, using Poincare-Linstedt perturbation theory and the GALI method.We show that these are connected with “weakly chaotic” regimes dominated bylow-dimensional tori, whose energy profile is exponentially localized in Fourierspace. We then study, at higher energies, diffusion phenomena and the transitionto delocalization and equipartition of the total energy among the N normal modes.

Then, in Chap. 7 we discuss different types of localization and diffusion phe-nomena this time in the configuration space of one-dimensional nonlinear lattices(chains of anharmonic oscillators). In particular, we start with the case where alllattice parameters are equal and show how exponentially localized oscillationscalled discrete breathers arise. These SPOs, which are often linearly stable andprovide a barrier to energy transport, can in fact be constructed by methods involvinginvariant manifold intersections that are described in Chap. 2. However, whentranslational invariance is broken by introducing random disorder in the parametersof the system, nonlinearity has a delocalizing effect and very important diffusivephenomena are observed, which persist for extremely long times.

In Chap. 8, we turn to the investigation of the type of statistics that characterizesthe dynamics in “weakly chaotic” regimes, where slow diffusion processes areobserved. Perhaps not surprisingly, we find that, as t grows, the probabilitydistribution functions (pdfs) associated with sums of chaotic variables in thesedomains do not quickly tend to a Gaussian at equilibrium, as expected by BGstatistical mechanics. Rather, they go through a sequence of quasi-stationary states(QSS), which are well-approximated by a family of q-Gaussian functions and sharesome remarkable properties in many examples of multi-dimensional Hamiltoniansystems. In these examples, it appears that “weakly chaotic” dynamics is connectedwith “stickiness” phenomena near the boundary of islands of regular motion called“edge of chaos”, where LCEs are very small and orbits get trapped for extremelylong times. Finally, Chap. 9 presents our conclusions, a list of open researchproblems and a number of other promising directions not treated in this book, whichare expected to shed new light on the fascinating and important subject of complexHamiltonian dynamics.

10 1 Introduction

1.4 Complex Hamiltonian Dynamics

In attempting to understand the complexity of chaotic behavior in Hamiltoniansystems, it is important to recall some basic facts about equilibrium thermodynamicsin the framework of BG theory of statistical ensembles. These topics form thebackbone of the results discussed in Chap. 8, but it is instructive to briefly reviewthem here. As is well-known in BG statistics, if a system can be at any one ofi D 1; 2; : : : ; W states with probability pi , its entropy is given by the famousformula

SBG D �k

WX

iD1

pi ln pi ; (1.21)

where k is Boltzmann’s constant, provided, of course,

WX

iD1

pi D 1: (1.22)

The BG entropy satisfies the property of additivity, i.e. if A and B are twoindependent systems, the probability to be in their union is pACB

i;j D pAi pB

j andthis necessitates that the entropy of the joint state be additive, i.e.

SBG.A C B/ D SBG.A/ C SBG.B/: (1.23)

At thermal equilibrium, the probabilities that optimize the BG entropy subject to(1.22), given the energy spectrum Ei and temperature T of these states are:

pi D e�ˇEi

ZBG

; ZBG DWX

iD1

e�ˇEi ; (1.24)

where ˇ D 1=kT and ZBG is the so-called BG partition function. For a continuumset of states depending on one variable, x, the optimal pdf for BG statistics subjectto (1.22), zero mean and a given variance V is, of course, the well-known Gaussianp.x/ D e�x2=2V =

p2V . Another important property of the BG entropy is that it is

extensive, i.e.

limN !1SBG

N< 1: (1.25)

This means that the BG entropy grows linearly as a function of the number of dofN of the system. But then, what about many physically important systems that arenot extensive?

As we discuss in Chap. 8, it is for these type of situations that a different form ofentropy has been proposed [332], the so-called Tsallis entropy

1.4 Complex Hamiltonian Dynamics 11

Sq D k1 �PW

iD1 pqi

q � 1with

WX

iD1

pi D 1 (1.26)

depending on an index q, where i D 1; : : : ; W counts the states of the systemoccurring with probability pi and k is the Boltzmann constant. Just as the Gaussiandistribution represents an extremal of the BG entropy (1.21) for a continuum set ofstates x 2 R, the q-Gaussian distribution

P.x/ D ae�ˇx2

q a�1 � .1 � q/ˇx2/

� 11�q ; (1.27)

is obtained by optimizing the Tsallis entropy (1.26), where q is the entropic index,ˇ is a free parameter and a a normalization constant. Expression (1.27) is a

generalization of the Gaussian, since in the limit q ! 1 we have limq!1 e�ˇx2

q De�ˇx2

.The Tsallis entropy is, in general, not additive, since it can be shown that

Sq.ACB/ D Sq.A/CSq.B/Ck.1�q/Sq.A/Sq.B/ and hence is also not extensive.It thus offers us the possibility of studying cases where different subsystems likeA and B above are never completely uncorrelated, as is the case e.g. with manyrealistic physical systems where long range forces are involved [332]. Furthermore,as we argue in Chap. 8, the persistence of non-Gaussian QSS in “weakly chaotic“regimes of Hamiltonian systems represents another important case of highlycorrelated dynamics that must be more deeply understood. It appears, therefore, thata framework formally analogous to BG thermodynamics is needed for nonextensivesystems as well, where the exponentials in (1.24) are replaced by q-exponentials, asdefined in (1.27).

In the present book, complex Hamiltonian dynamics does not only refer tothe non-Gaussian statistics of “weakly chaotic” motion. Indeed, one may arguethat the quasi-stationary character of regular or ordered motion, as describedfor example in the analysis of FPU recurrences presented in Chap. 6, constitutesanother complicated phenomenon. More generally, our thesis is that complexity inHamiltonian systems lies in the transition from local to global behavior, exactlywhere one would expect it to be, you might say. The point we are trying to makeis that this is not merely a question of mathematical interest. As is by now widelyrecognized, it is precisely in this type of transition that important phenomena ofgreat physical relevance are found that need to be further explored.

Complexity is, of course, difficult to describe by a single definition. Broadlyspeaking, it characterizes systems of nonlinearly interacting components that canexhibit collective phenomena like pattern formation, self-organization and theemergence of states that are not observed at the level of any one individualcomponent. Clearly, therefore, Hamiltonian systems are not complex in the sensethere exists a solid mathematical framework for their accurate description. We donot need to invent new and inspiring models that will capture the main features ofthe dynamics, as we do in the flocking of birds, for example, or the spread of anepidemic.

12 1 Introduction

In Hamiltonian dynamics we shall speak of complexity when new phenomenaarise, which are not expected by the classical theory of Newtonian mechanics,or statistical physics. Our claim is that such phenomena do occur, for example,in regimes where different hierarchies of chaotic behavior are detected, wherephysical processes like diffusion and energy transport crucially depend on the rate ofseparation of nearby trajectories as a function of time. Even if this separation growsexponentially, it may do so in ways that are indistinguishable from a power-law dueto strong correlations present for very long times. In fact, it appears that this kind ofevolution is connected with the intricacies of a geometric complexity of phase spacedynamics caused by the presence of invariant Cantor sets located at the boundariesof islands of regular motion around stable periodic solutions of the system.

But this does not exhaust the sources of complexity in Hamiltonian dynamics.As we will discover, besides varying degrees of chaos, there also exist hierarchiesof order in N dof Hamiltonian systems. Indeed, around discrete breathers andnear NNMs responsible for FPU recurrences at low energies, one often finds thatinvariant tori exist of dimension much lower than the number N expected byclassical theorems. As we show in this book, one way to approach this issue is tostudy the stability of tori, using for example the GALI method, or group theoreticaltechniques in cases where the system possesses discrete symmetries. Much work,however, still needs to be done before these questions are completely understood.

What other complex phenomena are still waiting to be discovered in Hamiltoniandynamics? No one knows. If we judge, however, by the progress that has beenachieved to date and the rapidly growing interest of theoretical and experimentalresearchers, we would not be surprised if new results came to the surface, whoseimportance may rival analogous discoveries in non-Hamiltonian systems. Thechallenge is before us and the future of complex Hamiltonian dynamics looks verybright indeed.

Chapter 2Hamiltonian Systems of Few Degreesof Freedom

Abstract In Chap. 2 we provide first an elementary introduction to some simpleexamples of Hamiltonian systems of one and two degrees of freedom. We describethe essential features of phase space plots and focus on the concepts of periodic andquasiperiodic motions. We then address the questions of integrability and solvabilityof the equations, first for linear and then for nonlinear problems. We present theimportant integrability criteria of Painleve analysis in complex time and show, on thenon-integrable Henon-Heiles model, how chaotic orbits arise on a Poincare Surfaceof Section of the dynamics in phase space. Using the example of a periodicallydriven Duffing oscillator, we explain that chaos is connected with the intersection ofinvariant manifolds and describe how these intersections can be analytically studiedby the perturbation approach of Mel’nikov theory.

2.1 The Case of N D 1 Degree of Freedom

One of the first physical systems that an undergraduate student encounters in his(or her) science studies is the harmonic oscillator. This describes the oscillations ofa mass m, tied to a spring, which exerts on the mass a force that is proportional to thenegative of the displacement q of the mass from its equilibrium position (q D 0), asshown in Fig. 2.1. The dynamics is described by Newton’s second order differentialequation

md2q

dt2D �kq; (2.1)

where k > 0 is a constant representing the hardness (or softness) of the spring.Equation 2.1 can be easily solved by standard techniques of linear ODEs to yieldthe displacement q.t/ as an oscillatory function of time of the form

q.t/ D A sin.!t C ˛/; ! D pk=m; (2.2)


13

14 2 Hamiltonian Systems of Few Degrees of Freedom

Fig. 2.1 A harmonic oscillator consists of a mass m moving horizontally on a frictionless tableunder the action of a force which is proportional to the negative of the displacement q.t/ of themass from its equilibrium position at q D 0 (k > 0 is a proportionality constant)

Fig. 2.2 Plot of the solutionsof the harmonic oscillatorHamiltonian (2.4) in the.q; p/ phase space, as aone-parameter family ofcurves, for different values ofthe total energy E

where A and ˛ are free constants corresponding to the amplitude and phase ofoscillations and ! is the frequency.

If we now recall from introductory physics that p D mdq=dt represents themomentum of the mass m, we can rewrite (2.1) in the form of two first order ODEs

dq

dtD p

mD @H

@p;

dp

dtD �kq D �@H

@q; (2.3)

where we have already anticipated that the dynamics is derived from the Hamilto-nian function

H.q; p/ D p2

2mC k

q2

2D E; (2.4)

which is an integral of the motion (since dH=dt D 0), whose value E represents thetotal (kinetic plus potential) energy of the system. Thus, this mass-spring system isour first example of a Hamiltonian system of N D 1 dof.

We will not attempt to solve (2.3). Had we done so, of course, we would haverecovered (2.2). Instead, we will consider the solutions of the problem geometricallyas a one-parameter family of trajectories (or orbits) given by (2.3) and plotted inthe .q; p/ phase space of the system in Fig. 2.2. Thus, as this figure shows, bothvariables q.t/ and p.t/ oscillate periodically with the same frequency !. Theirorbits are ellipses whose (semi-) major and minor axes are determined by theparameters k, m and the value of E .

Note now that the energy integral (2.4) can also be used to bring us one stepcloser to the complete solution of the problem, by solving (2.4) for p and obtainingan equation where the variable q is separated from the time t . Integrating both sidesof this equation in terms of the respective variables, we arrive at the expression

2.1 The Case of N D 1 Degree of Freedom 15

Z q

0

dqq

2E=m � !2q2

D t C c; (2.5)

which defines implicitly the solution q.t/ as function of t , c being the second freeconstant of the problem (the first one is E). The reader now recognizes the left sideof (2.5) as the inverse sine function, whence we finally obtain

q.t/ Dr

2E

ksin.!.t C c//; (2.6)

where the constants E and c can be related to A and ˛, by direct comparison with(2.2).

The process described above is called integration by quadratures and was madepossible by the crucial existence of the integral of motion (2.4).

Of course, the harmonic oscillator is a linear system whose dynamics is verysimple. It possesses a single equilibrium or fixed point at .0; 0/, where all firstderivatives in (2.3) vanish. This point is called elliptic and all closed curves about itin Fig. 2.2 describe oscillations whose frequency ! is independent of the choice ofinitial conditions.

Let us turn, therefore, to a more interesting one dof Hamiltonian systemrepresenting the motion of a simple pendulum shown in Fig. 2.3. Its equation ofmotion according to classical mechanics is

ml2 d2�

dt2D �mlg sin �; (2.7)

where the right side of (2.7) expresses the restoring torque due to the weight mg

of the mass (g being the acceleration due to gravity) and the left side describes thetime derivative of the angular momentum of the mass. If we now write this equationas a system of two first order ODEs, as we did for the harmonic oscillator, we findagain that they can be cast in Hamiltonian form

dq

dtD p D @H

@p;

dp

dtD �g

lsin q D �@H

@q; (2.8)

Fig. 2.3 The simplependulum is moving in thetwo-dimensional plane .x; z/,but its motion is completelydescribed by the angle � andits derivative d�=dt


Fig. 2.4 Plot of the curvesof the energy integral (2.9)in the .q; p/ phase space.Note, the presence ofadditional equilibrium pointsat .˙�; 0/, which are of thesaddle type. Observe also thecurve S separating oscillatorymotion (L) about the centralelliptic point at .0; 0/ fromrotational motion (R)

where q D � and l is the length of the pendulum. Of course, we cannot solvethese equations as easily as we did for the harmonic oscillator. Still, as youwill find out by solving Exercise 2.1, the solution of (2.8) can be obtained viathe so-called Jacobi elliptic functions [6, 107, 169], which have been thoroughlystudied in the literature and are directly related to the solution of an anharmonicoscillator with cubic nonlinearity, about which we will have a lot to say in latersections.

Still, we can make great progress in understanding the dynamics of the simplependulum through its Hamiltonian function, which provides the energy integral

H.q; p/ D p2

2C g

l.1 � cos q/ D E: (2.9)

Plotting this family of curves in the .q; p/ phase space for different values of E

we now obtain a much more interesting picture than Fig. 2.2 depicted in Fig. 2.4.Observe that, besides the elliptic fixed point at the origin, there are two newequilibria located at the points .˙�; 0/. These are called saddle points and repelmost orbits in their neighborhood along hyperbola-looking trajectories. For thisreason points A and B in Fig. 2.4 are also called unstable, in contrast to the .0; 0/

fixed point, which is called stable. More precisely, in view of the theory presented inChap. 1, .0; 0/ is characterized by what we called conditional (or neutral) stability.

Let us now discuss the linear stability of fixed points of a Hamiltonian systemwithin the more general framework outlined in Sect. 1.2. According to this analysis,we first need to linearize Hamilton’s system of ODEs (1.18) about a fixed point,which we assume to be located at the origin 0 D .0; 0; : : : ; 0/ of the 2N D phasespace. The resulting set of linear ODEs is written in terms of the Jacobian matrixJ D ˚

Jjk

�:

Pxj D2NX

kD1

Jjkxk; j D 1; 2; : : : ; 2N; (2.10)

whose elements (show this as a simple exercise starting with (1.18))

2.1 The Case of N D 1 Degree of Freedom 17

j D 1; 2; : : : ; N W Jjk D @2H

@pj @qk

.0/; k D 1; : : : ; N;

Jjk D @2H

@pj @pk�N

.0/; k D N C 1; : : : ; 2N;

j D N C 1; : : : ; 2N W Jjk D � @2H

@qj �N @qk

.0/; k D 1; : : : ; N;

Jjk D � @2H

@qj �N @pk�N

.0/; k D N C 1; : : : ; 2N;

(2.11)

where xj represents small variations away from the origin (qj D pj D 0, j D1; : : : ; N ) of the qj variables for j D 1; : : : ; N and the pj variables for j DN C 1; : : : ; 2N .

For Hamiltonian systems of N D 1 dof, it is clear from (2.11) that the elementsof the 2 � 2 Jacobian matrix J D ˚

Jjk

�; j; k D 1; 2 are

J11 D �J22 D @2H

@[email protected]; 0/; J12 D @2H

@p2.0; 0/; J21 D �@2H

@q2.0; 0/: (2.12)

In the case of the harmonic oscillator (2.3) this matrix has two imaginary eigen-values �1 D i! and �2 D �i!, and hence corresponds to the case of conditional(neutral) stability defined in Sect. 1.2. As a local property, it characterizes the fixedpoint at the origin of Fig. 2.2 as elliptic. However, since the original problemis linear, this type of stability is also global and implies that the solutions areoscillatory not only near the origin but in the full phase plane .q; p/.

In the case of the pendulum Hamiltonian, however, the situation is very different.First of all, there are more than one fixed points. In fact, there is an infinity of themlocated at .�n; 0/ D .˙n�; 0/; n D 0; 1; 2; : : :, where . Pq; Pp/ D .0; 0/ in the .q; p/

phase space. The linearized equations of motion about these fixed points are:

� Px1

Px2

�D�

J11 J12

J21 J22

��x1

x2

�

D�

0 1

� g

lcos �n 0

��x1

x2

�D�

0 1

� g

l.�1/n 0

��x1

x2

�: (2.13)

Thus, for n D 2k even (k any integer) the Jacobian matrix J has two complexconjugate (imaginary) eigenvalues �1 D i!0, �2 D �i!0 and corresponds to ellipticfixed points at .2k�; 0/, while for n D 2k C 1, the equilibria are of the saddletype with real eigenvalues �1 D !0, �2 D �!0, where !0 D p

g=l (see pointsA, B in Fig. 2.4). Near these saddle points, the motion is governed by the two realeigenvectors corresponding to the above two eigenvalues. Along the eigenvector


corresponding to �1 > 0, solutions of (2.13) move exponentially away from thefixed point, while along the �2 < 0 eigenvector they exponentially converge to thefixed point.

The important observation here is that, in the case of the simple pendulum, thesolutions of the linearized equations of motion (2.13) are valid only locally within aninfinitesimally small neighborhood of the fixed points. The reason for this is that thefull equations of the system (2.8) are nonlinear and hence there is no guarantee thatthe linear dynamics near the fixed points is topologically equivalent to the solutionsof the system when the nonlinear terms are included.

By topological equivalence, we are referring to the existence of a homeomor-phism (i.e. a continuous and invertible map, with a continuous inverse) that mapsorbits of (2.13) in a one-to-one way to orbits of (2.8), see [170, 265, 346]. How dowe know that? How can we be sure that if we expand the right hand sides of (2.8)in powers of q, p, using e.g. sin q D q � q3=6 C : : : as we did in Sect. 1.2, wewill obtain solutions which are topologically equivalent to those of the linearizedequations (2.13)?

The answer to this question is provided by the Hartman-Grobman theorem[170, 175, 257, 265, 346], which states that such an equivalence can be establishedprovided all eigenvalues of the Jacobian matrix have real part different from zero.Now, as we learned from Lyapunov’s theory, if all eigenvalues satisfy Re.�i / < 0,the equilibrium is asymptotically stable. However, if there is at least one eigenvaluewhose real part is positive, the general solutions of the nonlinear system deviateexponentially away from the fixed point.

Well, this settles the issue about the saddle points at ..2k C 1/�; 0/ of thesimple pendulum. It does not help us, however, with the elliptic points at .2k�; 0/,since the linearized equations about them have Re.�i / D 0, i D 1; 2 and theHartman-Grobman theorem does not apply. Here is where the integral of the motionrepresented by the Hamiltonian function (2.9) comes to the rescue. It establishesthe existence of a family of closed curves globally, i.e. within a large region aroundthese elliptic points.

In our problem, this region extends all the way to the separatrix curve S joiningthe unstable manifold of saddle point A to the stable manifold of saddle point B (andthe unstable manifold of B with the stable one of A), as shown in Fig. 2.4. Thesemanifolds are analytic curves which are tangent to the corresponding eigenvectorsof the linearized equations and are called invariant, since all solutions starting onsuch a manifold remain on it for all t > 0 (or t < 0).

Invariant manifolds, however, do not need to join smoothly to form separatrices,as in Fig. 2.4. In fact, as we explain in the next sections, invariant manifolds ofHamiltonian systems of N � 2 dof generically intersect each other transversally(i.e. with non-zero angle) and give rise to chaotic regions in the 2N D phase spaceR

2N . If, on the other hand, these manifolds do join smoothly, this suggests that thesystem has N analytic, single-valued integrals and is hence completely integrablein the sense of the LA theorem, as in the case of the simple pendulum.

2.2 The Case of N D 2 Degrees of Freedom 19

2.2 The Case of N D 2 Degrees of Freedom

Following the above discussion, it would be natural to extend our study toHamiltonian systems of two dof, joining at first two harmonic oscillators, as shownin Fig. 2.5 and applying the approach of Sect. 2.1. We furthermore assume that ouroscillators have equal masses m1 D m2 D m and spring constants k1 D k2 D k andimpose fixed boundary conditions to their endpoints, as shown in Fig. 2.5. Clearly,Newton’s equations of motion give in this case:

md2q1

dt2D �kq1 � k.q1 � q2/; m

d2q2

dt2D �kq2 C k.q1 � q2/; (2.14)

where qi .t/ are the particles’ displacements from their equilibrium positions atqi D 0, i D 1; 2. If we also introduce the momenta pi .t/ of the two particles interms of their velocities, as we did for the single harmonic oscillator, we arrive atthe following fourth order system of ODEs

dq1

dtD p1

m;

dp1

dt�kq1�k.q1�q2/;

dq2

dtD p2

m;

dp2

dt�kq2Ck.q1 �q2/; (2.15)

which is definitely Hamiltonian, since it is derived from the Hamiltonian function

H.q1; p1; q2; p2/ D p21

2mC p2

2

2mC k

q21

2C k

q22

2C k

.q1 � q2/2

2D E (2.16)

(see (1.18)), expressing the integral of total energy E . Note that we may think of thetwo endpoints of the system as represented by positions and momenta q0, p0 and q3,p3 that vanish identically for all t .

What can we say about the solutions of this system? Are they all periodic, as inthe case of the single harmonic oscillator? Is the dynamics more complicated in thefour-dimensional phase space of the qi .t/, pi .t/, i D 1; 2 and in what way? Can westill use the integral (2.16) to solve the system by quadratures? Note, first of all, that,once we specify a value for the total energy E , our variables qi .t/, pi .t/, i D 1; 2

are no longer independent, as the motion actually evolves on a three-dimensionalso-called energy surface � .E/. If we choose to solve (2.16) for one of them, say

Fig. 2.5 Two coupled harmonic oscillators of equal mass m and force constant k, with displace-ments q1.t/, q2.t/ from their equilibrium position, moving horizontally on a frictionless table withtheir endpoints firmly attached to two immovable walls


p1, in terms of the other three, we do not arrive immediately at an equation whosevariables can be separated and thus integration by quadratures is impossible. Whatwould happen, however, if a second integral of the motion were available? Couldthe method of quadratures lead us to the solution in that case?

Your immediate reaction, of course, would be to say no. Two integrals could onlyserve to express one variable, say p1 again, in terms of two others, say q1 and q2,suggesting again that integration by quadratures is impossible. But, here is whereappearances deceive. Note that the difficulty arises because the equations of motion(2.14), as given to us by Newtonian mechanics, are coupled in the variables q1, q2.Is it perhaps possible to use a suitable coordinate transformation to separate themand write our equations as a system of two uncoupled harmonic oscillators?

2.2.1 Coordinate Transformations and Solutionby Quadratures

It is not very difficult to answer this question. Indeed, if we perform the change ofvariables

Q1 D q1 C q2p2

; Q2 D q1 � q2p2

; P1 D p1 C p2p2

; P2 D p1 � p2p2

(2.17)

(the factor 1=p

2 will be explained later), we see that, adding and subtracting bysides the two equations in (2.14) (dividing also by m and introducing ! D p

k=m),splits the problem to two harmonic oscillators

dQi

dtD Pi

m;

dPi

dtD �!2

i Qi ; i D 1; 2; !1 D !; !2 D p3!; (2.18)

which have different frequencies !1, !2. Observe that the new Hamiltonian of thesystem becomes

K.Q1; P1; Q2; P2/ D P 21

2mC P 2

2

2mC k

Q21

2C 3k

Q22

2D E: (2.19)

We now see that the factor 1=p

2 was introduced in (2.17) to make the newequations of motion (2.18) have the same canonical form as two uncoupledharmonic oscillators, while the new Hamiltonian K is expressed as the sum of theHamiltonians of these oscillators. In this way, we may say that changing from qi , pi

to Qi , Pi variables we have performed a canonical coordinate transformation to oursystem.

In this framework, we immediately realize that our problem possesses not onebut two integrals of the motion which may be thought of as the energies of the twooscillators


Fi .Q1; P1; Q2; P2// D P 2i

2mC ki

Q2i

2D Ei ; i D 1; 2; (2.20)

with k1 D k, k2 D 3k, while Ei are two free parameters of the system to be fixedby the initial conditions qi.0/, pi.0/, i D 1; 2. Thus, we may now proceed to applythe method of quadratures to each of these oscillators, as described in Sect. 2.1 toobtain the general solution of the system in the form

Qi .t/ Ds

2Ei

ki

sin.!i .t C ci //; i D 1; 2; (2.21)

ci being the other two free parameters needed for the complete solution of the fourfirst order ODEs (2.18). Naturally, if we wish to write our general solution in termsof the original variables of the problem, we only need to invert (2.17) to find q1.t/,q2.t/. We also remark that the above considerations show that the only fixed pointof the system lies at the origin of phase space and is elliptic (why?).

Let us make a crucial observation at this point: Note that the successfulapplication of the above analysis rests on the existence of two integrals of motion(2.20) associated with the free constants E1, E2 connected with the amplitudesof the corresponding oscillations. This shows that, for a system of two dof, twointegrals are necessary and sufficient for its integration by quadratures, as theremaining two arbitrary constants c1, c2 merely represent phases of oscillation andare not as important as E1 and E2. To understand this better, note that c1, c2 are notsingle-valued and hence do not belong to the class of integrals required by the LAtheorem for complete integrability.

All this suggests that it would be advisable to introduce one more transformationto the so-called action-angle variables Ii ; �i ; i D 1; 2 as follows

Qi Ds

2!iIi

ki

sin �i ; Pi Dp

2m!iIi cos �i ; i D 1; 2 (2.22)

and write our integrals (2.20) as Fi D Ii !i so that the new Hamiltonian (2.19) maybe expressed in a form

G.I1; �1; I2; �2/ D G.I1; I2/ D I1!1 C I2!2 (2.23)

that is independent of the angles �i , which also demonstrates the irrelevance of thephases of the oscillations in (2.21). Evidently, our action-angle variables also satisfyHamilton’s equations of motion

d�i

dtD @G

@Ii

D !i ;dIi

dtD �@G

@�i

D 0; (2.24)


which imply that the new momenta Ii are constants of the motion, while the �i areimmediately integrated to give �i D !i t C �i0, where �i0 are the two initial phases.Thus, the change to action-angle variables defines a canonical transformation andthe oscillations (2.22) are called linear normal modes of the system.

Let us now discuss the solutions of this coupled system of linear oscillators.Using (2.17) and (2.21) we see that they are expressed, in general, as linearcombinations of two trigonometric oscillations with frequencies !1 D p

k, !2 Dp3k. If these frequencies were rationally dependent, i.e. if their ratio were a rational

number !1=!2 D m1=m2 (m1; m2 being positive integers with no common divisor)all orbits would close on two-dimensional invariant tori, like the one shown inFig. 2.6 and the motion would be periodic (what are the m1; m2 of the orbit shownin Fig. 2.6?). Note that in our example, this could only happen for initial conditionssuch that E1 or E2 is zero, whence the solutions would execute in-phase or out-of-phase oscillations with frequency !2, or !1 respectively.

In the general case, though, E1 and E2 are both non-zero and the oscillations arequasiperiodic, in the sense that they result from the superposition of trigonometricterms whose frequencies are rationally independent, as the ratio !2=!1 D p

3

is an irrational number. Hence, the orbits produced by these solutions in the 4-dimensional phase space are never closed (periodic). Unlike the orbit shown inFig. 2.6, they never pass by the same point, covering eventually uniformly thetwo-dimensional torus of Fig. 2.6 specified by the values of E1 and E2.

Thus, in the spirit of the LA theorem we have verified for a Hamiltonian systemof two coupled harmonic oscillators that two global, single-valued, analytic integralsare necessary and sufficient to completely integrate the equations of motion byquadratures and moreover that the general solutions lie on two-dimensional tori,since the motion on the four-dimensional phase space of the problem is boundedabout an elliptic fixed point at the origin qi D pi D 0, i D 1; 2.

Let us now turn to the case of a system of two coupled nonlinear oscillators. Wecould choose to do so by introducing nonlinear terms in the spring forces involvedin the above two linear oscillators. However, we prefer to postpone such a studyfor later chapters, where we will deal extensively with one-dimensional chains ofN > 2 coupled nonlinear oscillators. Thus, we shall discuss here one of the mostfamous Hamiltonian two dof system, originally due to Henon and Heiles [164]

Fig. 2.6 If the ratio of the frequencies of a two dof integrable Hamiltonian system is a rationalnumber, the orbits are periodic (and hence close upon themselves) on two-dimensional invarianttori, such as the one shown in the figure. However, if the ratio !1=!2 is irrational, the orbits neverclose and eventually cover uniformly two-dimensional tori in the four-dimensional phase space ofthe coordinates qi , pi , i D 1; 2


H D 1

2m.p2

1 C p22/ C 1

2.!2

1 q21 C !2

2 q22/ C q2

1q2 � C

3q3

2: (2.25)

This describes the motion of a star of mass m in the axisymmetric potential of agalaxy whose lowest order (quadratic) terms are those of two uncoupled harmonicoscillators and the only higher order terms present are cubic in the q1, q2 variables(for a most informative recent review on the applications of nonlinear dynamics andchaos to galaxy models see [99]). Note that the value of m in (2.25) can be easilyscaled to unity by appropriately rescaling time, while an additional parameter beforethe q2

1q2 term in (2.25) has already been removed by a similar change of scale of allvariables qi , pi .

Thus, introducing the more convenient variables q1 D x, q2 D y, p1 D px ,p2 D py , we can rewrite the above Hamiltonian in the form

H D 1

2.p2

x C p2y/ C V.x; y/ D 1

2.p2

x C p2y/ C 1

2.Ax2 C By2/ C x2y � C

3y3 D E;

(2.26)where E is the total energy and we have set !2

1 D A > 0 and !22 D B > 0. It is

instructive to write down Newton’s equations of motion associated with this system

d2x

dt2D �@V

@xD �Ax � 2xy;

d2y

dt2D �@V

@yD �By � x2 C Cy2: (2.27)

Before going further, let us first write down here all possible fixed points of thesystem, which are easily found by setting the right hand sides in (2.27) equal tozero:

.i/ .x; y/ D .0; 0/; .ii/ .x; y/ D

0;

rB

C

!

;

.iii/ .x; y/ D

˙p

2BA C CA2

2; �A

2

! (2.28)

(the momentum coordinates corresponding to all these points are, of course, zero).The local (or linear) stability of these equilibria will be discussed later, withreference to some very important special cases of the Henon-Heiles problem.However, there is one of them whose character can be immediately deduced fromthe above equations: It is, of course, the origin (i), where all nonlinear terms in(2.27) do not contribute to the analysis. Thus, infinitesimally close to this point, theequations become identical to those of two uncoupled harmonic oscillator (sinceA > 0, B > 0) and therefore the equilibrium at the origin is elliptic.

Let us remark now that (2.26) represents a first integral of this system. If wecould also find a second one, as in the problem of the two harmonic oscillators,with all the nice mathematical properties required by the LA theorem, the problemwould be completely integrable and we would be able to integrate the (2.27) byquadratures and obtain the general solution. Unfortunately, in nonlinear two dof


Hamiltonian systems, the existence of such an integral is far from guaranteed. Infact, all indications for the Henon-Heiles system is that a second integral for all A,B , C does not exist.

“How does one know that?”, you may rightfully ask. Well, this is a very goodquestion. A first answer may be given by the fact that in many cases that have beenintegrated numerically by many authors, even close to the equilibrium point (i), onefinds, besides periodic and quasiperiodic orbits, a new kind of solution that appears“irregular” and “unpredictable”, one may even say random-looking [226]! These arethe orbits we have called chaotic. They tend to occupy densely three-dimensionalregions in the four-dimensional phase space and depend very sensitively on initialconditions, in the sense that starting at almost any other point in their immediatevicinity produces another trajectory (also chaotic), which deviates exponentiallyfrom the original (reference) orbit as time increases.

This would not happen, of course, if the problem was completely integrable,since in that case—at least close to the elliptic equilibrium point at the origin—all phase space would be occupied by two-dimensional tori, which are uniformlycovered by periodic or quasiperiodic solutions. These have nothing “irregular” or“unpredictable” about them. In fact, it can be shown that the distance betweennearby orbits in their vicinity deviate linearly from each other in phase space [226].

Still, if one works only numerically, it is possible to miss chaotic orbitsaltogether, since their associated regions can be very thin and may not evenappear in the computations. To speak about complete integrability therefore(or the absence thereof) one must have some analytical theory to support rigorouslyone’s statements. One such theory is provided by the so-called Painleve analysis,which investigates a very fundamental property of the solutions related to theirsingularities in complex time [98, 152, 279].

2.2.2 Integrability and Solvability of the Equations of Motion

It is of course very difficult to look at a pair of nonlinear coupled second orderODEs like (2.27) and try to integrate them by some clever change of variables, aswe did in the case of two coupled linear oscillators (2.14). Still, the careful readerwill notice (by inspection, as we say) that there is one obvious case where (2.27)can be directly integrated: A D B , C D �1. Under this condition, a transformationto sum and difference variables reduces the Henon-Heiles equations to the simpleuncoupled form

d2X

dt2D �AX � p

2X2;d2Y

dt2D �AY C p

2Y 2; X D x C yp2

; Y D x � yp2

(2.29)(see (2.17)).“This is great!”, you would exclaim: Now, we can multiply theseequations respectively by dX=dt and dY=dt and integrate them to arrive at thefollowing integrals of the motion


F1 D 1

2

�dX

dt

�2

C A

2X2 �

p2

3X3; F2 D 1

2

�dY

dt

�2

C A

2Y 2 C

p2

3Y 3: (2.30)

It is easy to see now that these two integrals are not independent from theHamiltonian (2.26). Indeed, if we also define the momentum coordinates PX DdX=dt , PY D dY=dt and observe that px D .PX CPY /=

p2, py D .PX �PY /=

p2

we easily verify that H D F1 C F2 and hence that the change to new variables is acanonical transformation.

Furthermore, it is possible to prove that the two integrals (2.30) are independentfrom each other. This can be done by demonstrating that the tangent space spannedby the partial derivatives of F1 with respect to X , PX and F2 with respect to Y , PY

at almost all points where F1 D F2 has the full dimensionality of the phase space ofthe system. Moreover, the Poisson bracket fF1; F2g vanishes which implies that thetwo integrals are in involution [19, 226]. We conclude, therefore, according to theLA theorem, that the system is completely integrable and hence its general solutioncan be obtained by quadratures.

Try it: If you go back to (2.30), solve for dX=dt and dY=dt and integrateby separation of variables, as we did for the two coupled harmonic oscillators,you will be led to elliptic integrals, which are the inverse of the so-called Jacobielliptic functions [6, 107, 169]. The most fundamental ones of them are sn.�t; �2/

and cn.�t; �2/ and their parameters �; � are related to the parameter A of ourproblem. When their so called modulus 0 � � � 1 tends to 1, the sn and cn tendto trigonometric functions (sine and cosine respectively), while � ! p

A, whichis the frequency of oscillations of our variables near the origin, see (2.29). On theother hand, when � ! 1, the period of sn and cn tends to infinity and they becomehyperbolic sinh and cosh functions (see [6, 107, 169] and Exercise 2.1).

So everything is fine with the A D B , C D �1 case of the Henon-Heilesequations. What do we do, however, for other choices of these parameters? Canwe still hope to come up with some “clever” transformations of variables, such asthe ones employed above, which will help us uncouple and eventually solve thenonlinear ODEs of the problem? The answer, unfortunately, is negative. As cleveras we may be, resourcefulness has its limits. Inspection alone is not sufficient tomake progress here. We need to develop a more systematic approach, such as theone provided by the so called Painleve analysis of (2.27) [98, 152, 279].

Let us, therefore, think more analytically and try to understand more about themathematics of the integrable case we discovered above. Our motivation is thefollowing: What if integrability (and hence also solvability) of our equations isconnected with the mathematical simplicity of the solutions of these equations? Inother words, what problems can we solve using the known mathematical functionsavailable in the literature today? And since we already discovered elliptic functionsin one example of our problem, why not try to find other integrable cases of theHenon-Heiles system, by requiring that their solutions belong to the same “class”as the Jacobi elliptic functions?


This is not such a crazy question. In fact, it was first asked by the famous Russianmathematician S. Kowalevskaya in the 1880s, who was thus able to discover onemore integrable case of the rotating body, in addition to the other three that hadalready been found by Euler and Jacobi. To this day, these are the only four knowncases, where the rotating body can be explicitly integrated. So the moral of the storyis clear: If you want to become famous, find a new integrable case of a physicallyimportant system of differential equations!

To understand what Kowalevskaya did we need to start from a remarkablemathematical property of the Jacobi elliptic functions: When their independentvariable u D �t is complex, their only singularities in the complex t-plane arepoles. In other words, these functions are not analytic everywhere in the complexdomain, and thus are more general than the ordinary exponentials and trigonometricfunctions of elementary mathematics. They are, however, single-valued, in the sensethat when evaluated on a closed loop around their singularities they recover theirvalues after every turn.

The recipe, therefore, appears rather simple: First express the solutions x.t/, y.t/

of (2.27) about a singularity at t D t� as series expansions that begin with a possiblepole of order r and s respectively. This gives

x.t/ D1X

kDr

ak�k y.t/ D1X

kDs

bk�k; � D t � t�; (2.31)

where r , s are integers and at least one of them is negative. These series are assumedto converge within a finite radius 0 < j� j D jt � t�j < in the complex t-plane andthus provide the complete solution of the problem near this singularity. It follows,therefore, that among the coefficients ak , bk three must be arbitrary, as t� is alreadyone of the free constants of the solution.

Substituting these series in (2.27) and balancing the most divergent terms leadsto the equations

r.r � 1/ar�r�2 D �2arbs�

rCs; s.s � 1/bs�s�2 D �a2

r �2r C Cb2s �2s ; (2.32)

which imply that there are two possible dominant behaviors (and hence two suchsingularity types) for this problem, namely

.i/ r D s D �2 W a�2 D ˙p18 C 9C ; b�2 D �3

.ii/ s D �2; r > s W ar D arbitrary; b�2 D r.1 � r/

2D 6

C:

(2.33)

Note that, in type (ii), the power r of the leading term of the expansion of x.t/

is directly related to the value of the parameter C . Since we have assumed thatour series involve only integer powers of � , the only C values that ensure this areC D �1; �2; �6. That’s great! The first one of them is already the beginning of


the thread that leads to the integrable case discovered earlier in this section by pureinspection.

What remains to be done now is entirely algorithmic and can even be pro-grammed on the computer: Take each one of the above cases and compute thecoefficients of the Laurent series of the solutions near the corresponding singularitytype. As we have already explained, at most three of the ak , bk should be arbitrary.And here is where the most important stumbling block of the method lies: Whenyou try to evaluate these coefficients you will discover that they satisfy a degeneratesystem of linear algebraic equations that is not homogeneous and hence is mostlikely incompatible for an arbitrary choice of the parameters A, B , C ! So what dowe do now?

Well, the first thing you can do is choose these parameter values in such a waythat these algebraic equations do become compatible. You will thus be rewarded byall the possible integrable cases that this procedure can identify. When all is saidand done, you will be pleasantly surprised to find that besides the case you alreadyknow about there are two new ones for which you can in principle now integratethe Henon-Heiles equations completely. Thus all the known integrable cases of theproblem can be summarized as follows:

Case 1 W A D B; C D �1;

Case 2 W A; B free; C D �6;

Case 3 W B D 16A; C D �16:

(2.34)

Well, to be completely honest, Case 3 does not arise automatically from the aboveprocedure. In fact, when you use C D �16 in the condition of singularity type (ii)you find that the value of the exponent r is a half integer. But that does not mean allis lost. It may be that if we switch from x to a variable u D x2, this new variablemay have a true Laurent series expansion with the required number of free constantsand that is precisely what happens here. Thus all three cases listed in (2.34) aboveare successfully identified by the Painleve analysis.

And what about C D �2? Well, in this case we are not so lucky. Althougheverything begins promisingly, we cannot find the required free constants unlesswe also introduce logarithmic terms in our series expansions. This means that thesolutions of the Henon-Heiles system for C D �2 contain logarithmic singularitiesalready at leading order and hence the above method does not apply. Indeed, whenwe integrate numerically the equations in this case we find that they possess chaoticsolutions and hence a second global, analytic and single-valued integral most likelydoes not exist [56].

In conclusion, we can say that if a system of ODEs has the so-called Painleveproperty, i.e. its solutions have only poles as movable singularities, it is expectedto be completely integrable, even explicitly solvable in terms of known functions.The term “movable” implies that the location of a singularity t� is one of thefree constants to be specified by the initial conditions of the problem. It serves


to differentiate movable singularities from the so-called fixed ones, which appearexplicitly in the equations of motion [107, 169].

Of course, identifying integrable cases by the Painleve analysis only tells uswhere to look for N dof Hamiltonian systems whose solutions are globally orderedand predictable. It does not tell us how to find the N integrals that must exist(according to the LA theorem) and which are necessary to solve the equations ofmotion by quadratures. But do we really want to get into all this trouble?

First of all the integrals are not at all easy to find. Even for the simple-lookingHenon-Heiles system , the best one can do is assume that the second integral F is apolynomial in the momentum and position variables and solve for its coefficients bydirect differentiation. What happens, however, if F turns out to be an infinite series?Well, in the Henon-Heiles we are lucky. The second integral in Cases 2 and 3 of(2.34) truncates to a polynomial of the form [60, 156]

Case 2 W F D x4 C 4x2y2 � 4p2xy C 4xpxpy C 4Ax2y

C .4A � B/.p2x C Ax2/;

Case 3 W F D 3p4x C 6.A C 2y/x2p2

x � 4x3pxpy � 4x4.Ay C y2/

C 3A2x4 � 2

3x6:

(2.35)

Now what? If we wish to arrive at the complete solution of the problem we mustsomehow use (2.35) to uncouple the variables of the problem and solve two one dofsystems as we did in integrable Case 1. But that is a far less trivial task in Cases 2and 3. So, again the question relentlessly arises: Why do it?

After all, integrable cases, you may argue, are so rare, that they can hardly beexpected to come up in realistic physical situations. Why worry about them? Whynot try to find out instead what happens to the Henon-Heiles for all other A, B , C

values apart from the ones of (2.34)? Well, don’t be impatient, we can always dothat numerically, of course. Before resorting to this alternative, however, it is worthnoting that integrable systems are especially useful for two important reasons: First,they frequently arise as close approximations of many physically realistic problems(the solar system being the most famous example) and second, their basic dynamicalfeatures do not change very much under small perturbations, as the KAM theorem[19] assures us and as we know by now from an abundance of examples.

Let’s take for instance the best-studied case of the Henon-Heiles system, whichis both physically interesting and (most likely) non-integrable, A D B D 1, C D 1

[164, 226]

H D 1

2.p2

x C p2y/ C 1

2.x2 C y2/ C x2y � 1

3y3 D E: (2.36)

Let us select a value of the total energy small enough, within the interval 0 < E <

1=6, for which it is known that all solutions of (2.36) are bounded. To visualize thesesolutions, let us plot in Fig. 2.7 the intersections of the corresponding orbits with the


Fig. 2.7 Orbit intersections with the PSS .y; py/ of the solutions of the Henon-Heiles system withHamiltonian (2.36), at times tk , k D 1; 2; : : :, where x.tk/ D 0, px.tk/ > 0 and for fixed valuesof the energy E . Note in panels (a) and (b), where E D 1=24 and E D 1=12 respectively, nochaotic orbits are visible and a second integral of motion besides (2.36) appears to exist. Whenwe raise the energy however to E D 1=8 in (c), this is clearly seen to be an illusion. Ordered(i.e. quasiperiodic) motion is restricted within islands surrounded by chaos, whose size diminishesrapidly as the energy increases further. Thus, in (d) where E attains the value E D 1=6 abovewhich orbits escape to infinity, the domain of chaotic motion extends over most of the availableenergy surface

plane .y; py/ every time t D tk , k D 1; 2; : : : at which x.tk/ D 0, px.tk/ > 0.This is what we call a Poincare surface of section (PSS) for which we shall havea lot more to say in the remainder of this chapter. Let us not forget, after all, thatit was Poincare himself who first pointed out the complexity of the solutions ofHamiltonian systems such as (2.36) in his celebrated study of the non-integrabilityof the three-body problem of celestial mechanics [274].

Do you see any chaos in panel (a) of this figure, where E D 1=24? No. Yet,it is there, between the invariant curves that correspond to intersections of two-dimensional tori of quasiperiodic orbits, in the form of thin chaotic layers that aretoo small to be visible. To see these chaotic solutions, one would have to eitherincrease the resolution of Fig. 2.7a or raise the energy. Let us do the latter. Observein this way, in Fig. 2.7b, c, that we would have to reach energy values as high as


E D 1=8 before we begin to see widespread chaotic regions on the PSS. Note alsothat these regions grow rapidly as E approaches the escape energy E D 1=6, wherechaos seems to spread over the full domain of allowed motion, while regular orbitsare restricted within small “islands” of quasiperiodic motion that seem to vanish asthe energy increases.

How can we better understand these dynamical phenomena? What is the degreeof their complexity? Is Hamiltonian dynamics just a battle between order and chaosand all we need to do is find out who is “winning” by estimating their respectivedomains of “influence”? Well, even that would not be such a small accomplishment!Clearly, however, to make progress in this direction, we would have to know moreabout the “border” that separates order from chaos in Hamiltonian systems. And, asit often happens in mathematical physics, to understand a problem deeper, we needto analyze first in detail its simplest occurrences.

2.3 Non-autonomous One Degree of Freedom HamiltonianSystems

Perhaps the simplest non-integrable Hamiltonian system in existence is a singleperiodically driven anharmonic oscillator, studied by Georg Duffing [116], aGerman engineer who worked on nonlinear vibrations. This oscillator is describedby what is called nowadays Duffing’s equation of motion [160, 205, 346]

Rq.t/ C !20 q.t/ C ˛q2.t/ C ˇq3.t/ C sin ˝t D 0; (2.37)

where we denote from now on time differentiation by a “dot” over the differentiatedvariable. In (2.37) q.t/ again represents the displacement of this one-dimensionaloscillator from its zero position. Furthermore, we can also denote its momentum byp D Pq and write the above equation as a driven one dof Hamiltonian system of theform

Pq D p D @H

@p; Pp D �!2

0 q � ˛q2 � ˇq3 � sin ˝t D �@H

@q; (2.38)

where the Hamiltonian function

H D H.q; p; t/ D 1

2.p2 C !2

0 q2/ C ˛

3q3 C ˇ

4q4 C q sin ˝t; (2.39)

for ¤ 0 depends explicitly on t . Thus, H is no longer an integral of the motion.Indeed, when ¤ 0 things quickly start to get complicated. First of all, q D

p D 0 is no longer an equilibrium position since the sinusoidal term in (2.37) willimmediately drive the oscillator away from that position. In fact, no equilibria existat all, since it is clearly not possible to find any points q0, p0 that remain fixedwhen we set Pq D Pp D 0 in these equations. More importantly, of course, the phase

2.3 Non-autonomous One Degree of Freedom Hamiltonian Systems 31

space of the system is not two-dimensional. Observe that it is no longer sufficient tospecify a point .q.t0/; p.t0// in the .q; p/ plane and expect to get a unique solutionlying in that plane. You also need to specify exactly what time t0 you are talkingabout, since the dynamics is no longer invariant under time translation, as in thecases of the autonomous Hamiltonian systems considered so far.

The motion, therefore, evolves in the three-dimensional space, q; p; t and (2.39)is sometimes referred to as a system of one and a half dof. In fact it is no differentthan a two dof Hamiltonian system, since, in the terminology of Sect. 2.1, we canrepresent the nonlinear oscillator in (2.37) by one action-angle pair I1, �1 and thinkof q, p as coupled to a second (linear) oscillator whose action I2 is fixed, while itsangle coordinate is �2 D ˝t . Mathematically speaking, we say that the phase spaceis a cylinder, R2 � S1, where every point in the .q; p/ plane is associated with anangle on the unit circle S1.

To understand how this periodic driving in (2.37) affects the dynamics we willconsider the periodic diving term as a small perturbation in the two special casesˇ D 0 and ˛ D 0 separately.

2.3.1 The Duffing Oscillator with Quadratic Nonlinearity

Let us set ˇ D 0, ˛ D �1, !0 D 1 in (2.38) and examine the solutions of the system

Pq D p; Pp D �q C q2 � sin ˝t (2.40)

for 0 � � 1. When D 0 it is easy to see that there are two equilibria: (1)q D p D 0, (2) q D 1, p D 0. Linearizing the equations of motion about them,as described in Sect. 2.1, we easily find that (1) is a (stable) elliptic point and (2)is an (unstable) saddle. On the other hand, the dynamics of this one dof system inthe full phase plane .q; p/ is described by the family of curves H D H.q; p/ D.1=2/.p2 C q2/ � .1=3/q3 D E parameterized by the value of the energy E , asshown in Fig. 2.8a.

Observe that, as in the case of the simple pendulum, here also a region ofoscillations exists around the origin, which extends all the way to the saddle at .1; 0/

and is separated from the outside part of the phase plane (where all solutions escapeto infinity) by an invariant curve S called the separatrix. The only difference is thatin the pendulum problem S joins two different saddles and represents a so-calledheteroclinic solution of the equations, while in the quadratic oscillator of (2.40),S joins .1; 0/ to itself and is called homoclinic solution (these terms come fromthe Greek words “cline” meaning “bed” and “hetero” and “homo” which mean,of course, “different” and “same” respectively). These invariant curves, and thehomoclinic (or heteroclinic) orbits that lie on them, constitute perhaps the singlemost important object of study in nonlinear Hamiltonian dynamics.

Let us try to understand why: As Poincare first pointed out, when ¤ 0, theelliptic point at .0; 0/ and the saddle point at .1; 0/ in Fig. 2.8a shift to slightlydifferent locations P and Q on the PSS


Fig. 2.8 (a) The .q; p/ phase plane of the quadratic oscillator (2.40) in the D 0 case. Note theelliptic point at the origin and the saddle point at .1; 0/. The separatrix S here is a single invariantcurve described by a homoclinic orbit joining the unstable and stable manifolds of the saddle at.1; 0/. (b) When ¤ 0, however, these manifolds no longer join smoothly but intersect each otherinfinitely often on a PSS ˙t0 D .q.tk/; p.tk//, tk D kT C t0, k D 1; 2; : : :, where T D 2�=˝ isthe period of the forcing term

˙t0 D fq.tk/; p.tk//; tk D kT C t0; k D ˙1; ˙2; : : :g ; T D 2�=˝ (2.41)

(see Fig. 2.8b). These are no longer fixed points of the differential equations, buthave become intersections of T .D 2�=˝/-periodic orbits of the system (2.40),whose solutions evolve now in the three-dimensional space R

2 � S1, as explainedabove [160, 226].

In addition, the stable and unstable manifolds SC and S� of the point Q intersecttransversally at a point .q0; p0/ in Fig. 2.8b, lying furthest away from Q. Thisimplies the simultaneous occurrence of a double infinity of such points, denotedby .qk; pk/, k D ˙1; ˙2; : : :, with .qk; pk/ ! P as k ! ˙1, which constitutein fact two distinct homoclinic orbits, one with k D 0; ˙2; : : : ; ˙2m; : : : and onewith k D ˙1; : : : ; ˙.2m C 1/; : : :, m being a positive integer.

The reason for this is that the time evolution of the system on the PSS (2.41) ismore conveniently described by the Poincare map [170, 265, 346]

Pt0 W ˙t0 ! ˙t0 ; t0 2 .0; 2�=˝/; (2.42)

defined by following the trajectories of the systems and monitoring their intersec-tions with ˙t0 . P and Q are respectively an elliptic and a saddle fixed point of thismap, since they satisfy Pt0.P / D P and Pt0.Q/ D Q. Most importantly, the map(2.42) is a a diffeomorphism, i.e. it is continuous and at (least once) continuouslydifferentiable with respect to the phase space variables q, p. It is also invertible,i.e. P �1

t0exists and is also a diffeomorphism. This means that it maps, forward

and backward in time, neighborhoods of the PSS that exhibit diffeomorphicallyequivalent dynamics. Thus, an intersection point of the invariant manifolds ismapped by Pt0 (or P �1

t0) to the next (or previous) such intersection point, provided

the manifolds in their neighborhoods have the same orientation.


Fig. 2.9 (a) The successive action of (2.42) maps each shaded lobe of the figure to the next one,closer to the point Q, thus taking points from the region “inside” the invariant manifolds to theregion “outside” and eventually to infinity. (b) Schematic picture of the dynamics around theelliptic fixed point of the Poincare map (2.42), on the surface of section ˙t0 , when ¤ 0

This explains why, for example, as the homoclinic point X is mapped to X 0, X 00,etc. (and its companion Y is mapped to Y 0, Y 00, etc.) in Fig. 2.9a, a shaded lobeis mapped to the next one closer to the saddle point Q, by successive applicationsof the map Pt0 . This results in the transport of points which were once “inside”the manifolds SC, S� to the region “outside”. On the other hand, as Fig. 2.9a alsoshows, there are corresponding lobes which were originally “outside” and are nowmapped “inside” the domain of the elliptic point at P . Furthermore, the Hamiltoniannature of the problem implies, according to Liouville’s theorem [226,265,346], thatthe system is conservative in the sense that it preserves phase space volume. Theconsequence of this is that the areas of the lobes shown in Fig. 2.9 are equal to eachother.

Meanwhile, in the homoclinic tangle formed by the intersecting manifolds nearthe saddle point Q there is a wealth of dynamical phenomena about which a lotis known. Besides the homoclinic orbits, there is a countable infinity of unstableperiodic solutions, while there also exist invariant Cantor sets, for which it can berigorously proved by the Smale horseshoe theory [160,265,346] that any two nearbypoints belonging to this set lead to orbits that separate from each other exponentiallyfast in phase space.

Of course, if instead of one saddle point our Poincare map had two (see nextsubsection) the intersecting manifolds would give rise to heteroclinic orbits, inthe same way as described above, with lobes of equal areas transporting pointsfrom “inside” the invariant manifolds “outside” and vice versa. Heteroclinic tangleswould thus be formed, where invariant sets of chaotic orbits can be similarly provedto exist. The important observation is that this does not happen only near saddlefixed points of the Poincare map (2.42). It also happens, at smaller scales, nearevery unstable m-periodic orbit of the system of period Tm D mT , m D 2; 3; : : :,intersecting the PSS (2.41) at m points (see Fig. 2.9b).

As we know from the KAM theorem, “most” (in the sense of positive measure)invariant curves around the origin of (2.40) at D 0 are preserved for sufficientlysmall ¤ 0. Furthermore, a very important theorem by Birkhoff [226, 346] also


assures us that the invariant curves of the D 0 case corresponding to periodicorbits break up into an even number of stable and unstable m-periodic orbits, whichappear on the surface of section as elliptic and saddle fixed points respectively ofthe mth power of the Poincare map P m

t0.

This is schematically shown in Fig. 2.9b. Are you impressed? You should be. Thisis indeed a wonderful sign of a complex phenomenon, as it rigorously demonstratesthat there are chaotic regions around saddle points present at all scales! Aroundevery elliptic fixed point of any power of the Poincare map (hence around everysmall island shown in Fig. 2.9b), there are smaller islands and around them evensmaller ones etc., with saddles and tangles of heteroclinic chaos in between downto infinitesimally small scales!

Did you expect all this to happen, as soon as we allowed ¤ 0? Clearly, turningon the driving force in (2.40) was not such an innocent move. The Hamiltonianceased to be an integral and the dimensionality of the motion was increased to three,where an amazing variety of new phenomena are possible, the most important ofthem being the occurrence of chaotic orbits.

2.3.2 The Duffing Oscillator with Cubic Nonlinearity

Let us now repeat the above exercise for the Duffing oscillator (2.37) with onlycubic nonlinearity. Setting ˛ D 0, the equation of motion becomes

Rq.t/ D �ıq.t/ � ˇq3.t/ C sin ˝t; (2.43)

where we have replaced the parameter before the linear term by ı, so as to be ableto consider also the case ı < 0. Although the phenomena we encounter here arequalitatively the same as in the example of the quadratic Duffing oscillator, we willhave the chance to exploit the additional symmetry of the dynamics under reflectionabout the axis q D 0 (when D 0). This is relevant in many physical systems (likevibrating structures fixed on a horizontal uniform floor), where motions to the rightor left are equally favorable. With (2.43), we also have the opportunity to discuss acase that has many common features with the pendulum problem and examine theinteresting dynamics of the so-called double-well potential that arises frequently inmany problems of physics.

Let us begin by considering the case ı D �ˇ D 1, where the above system iswritten in q, p phase space coordinates as

Pq D p; Pp D �q C q3 C sin ˝t: (2.44)

Evidently, in the unforced case D 0, this anharmonic oscillator possesses threefixed points: (1) An elliptic one, P D .0; 0/ and two saddle points (2) Q1 D .�1; 0/

and (3) Q2 D .1; 0/. The phase space plot is thus very similar with that of thependulum (see Fig. 2.4), only in the case of the pendulum Q1 D .��; 0/, Q2 D


.�; 0/ and the picture is repeated about every elliptic point at .2m�; 0/ and its twosaddles at ..2m ˙ 1/�; 0/.

Note that our separatrices here represent heteroclinic orbits joining the saddlepoints at .˙1; 0/. It is not difficult to show (see Exercise 2.3) that, in the unforcedcase, both cubic and quadratic oscillators (2.40), (2.44), possess the Painleveproperty and are explicitly solvable in terms of Jacobi elliptic functions. This meansthat their solutions along these separatrices are given in terms of simple hyperbolicfunctions. We thus have the opportunity to apply to these solutions the so-calledMel’nikov theory [160,346], which allows us to study exactly what happens to theseheteroclinic orbits when ¤ 0. In Problem 2.3 we ask you to carry out such acalculation, following the steps outlined below.

First of all, recall that the saddle points of our cubic oscillator (just as in thecase of the quadratic oscillator) are unstable periodic orbits of (2.44) with periodT D 2�=˝ , which persist on the PSS (2.41) as saddle fixed points of the Poincaremap (2.42). These points continue to possess stable and unstable manifolds, whichno longer join smoothly, as in the unforced case. Mel’nikov’s approach workswithin the framework of perturbation theory for j j � 1 and yields expressionsfor the heteroclinic solutions of the perturbed problem, both for the stable as well asunstable manifolds of these saddle fixed points, as series expansions in powers of .

We thus obtain different expansions which are uniformly valid in time intervals.�1; t1/, for the unstable manifold of the Q1 point and .t2; 1/ for the stablemanifold of the point Q2, where t2 > t1. Examining these solutions at timest0 2 .t1; t2/ where both series are valid, Mel’nikov’s theory shows, to first orderin , that the points at which the corresponding manifolds intersect correspond tothe roots of the function

M.t0/ DZ 1

�1.f1g2 � f2g1/. Ox.t � t0/; t/dt; (2.45)

which is proportional to the distance between the two manifolds on the PSS (2.41),as a function of t0 (see Problem 2.3). To derive (2.45) we have written our equationsof motion in the form

Pq D f1.q; p/ C g1.q; p; t/; Pq D f2.q; p/ C g2.q; p; t/;

gi .q; p; t/ D gi .q; p; t C T /; (2.46)

i D 1; 2, while Ox.t � t0/ D . Oq.t � t0/; Op.t � t0// represents the exact heteroclinic(or homoclinic) solution of the unforced D 0 case.

This is fantastic! It means that we can now study analytically what happens tothese stable and unstable manifolds as one turns on the perturbation of the periodicforcing in the above systems. Indeed, based on our knowledge of the exact solutionon these manifolds for D 0 we can verify, as Poincare predicted, that the Melnikovintegral (2.45) is an oscillatory function of t0 with infinitely many roots, M.t0i / D 0,i D 0; ˙1; ˙2; : : :. These correspond to the intersection points of the manifolds atwhich the heteroclinic (or homoclinic) orbits of the perturbed system are located.


Let us see how all this works in the case of the cubic oscillator (2.44). As thereader can easily verify, for D 0, the separatrices (or heteroclinic solutions) ofthis problem, S˙, are given by the hyperbolic functions

Ox.t � t0/ D . Oq.t � t0/; Op.t � t0// D ˙�

tanh

�1p2

.t � t0/

�;

1p2

sech2.t � t0/

�:

(2.47)If we use the above expressions to substitute in (2.45) we obtain an integral

M.t0/ DZ 1

�1Op.t � t0/ sin ˝tdt D 1p

2

Z 1

�1sech2t sin ˝.t C t0/dt; (2.48)

that is not elementary. Its evaluation requires that we extend the integration to astrip in the complex t-plane and use Cauchy’s residue theorem (see Problem 2.3).The result is a simple formula

M.t0/ D �˝=2

sinh.�˝=2/sin ˝t0; (2.49)

which, when multiplied by and divided by the magnitude of the unperturbed

vector fieldq

f 21 .�/ C f 2

2 .�/ provides an estimate of the distance between thestable and unstable manifolds at t D t0 C � (compare with the numerical resultsof Exercise 2.4). The above formulas demonstrate that M.t0/ has infinitely manyzeros (due to the sine function in (2.49)), while the amplitude of its oscillationsgrows exponentially as � increases and we move closer to the saddle points Q1, Q2.

It is important to note that Mel’nikov’s theory does not require that the vectorfields Of D .f1; f2/ or Og D .g1; g2/ be Hamiltonian! All it demands is thatthe unperturbed system possesses a solution Ox.t/ that is either a homoclinic orheteroclinic orbit. In particular, if our Duffing oscillators involve a small dissipativeterm of the form ��p, say, with � > 0 of the same order as , we can include itin the periodic perturbation Og D .g1; g2/ and study its crucial role in inhibiting theintersection of the invariant manifolds. Indeed, it can be shown that each � providesa threshold that must exceed for Smale horseshoe chaos to occur in the system!To understand exactly how this important effect can be studied using Mel’nikov’stheory, the reader is encouraged to solve Problem 2.4.

In fact, Mel’nikov’s theory does not only apply to perturbations of planar Hamil-tonian systems. It can be extended to dynamical systems of arbitrary dimension,n � 2, which can be written in the form

Px.t/ D f.x.t// C g.x; t/; g.x; t/ D g.x; t C T /; (2.50)

for x.t/ D .x1.t/; : : : ; xn.t// and a sufficiently small real parameter. In theparticular case where the unperturbed equations are derived from an integrable


Hamiltonian system, possessing n independent integrals in involution Fi , i D1; 2; : : : ; n, we obtain a Mel’nikov vector function, M.t0; �/ D .M1; : : : ; Mn/,whose components are given by [90, 287, 288]

Mi .t0; �/ DZ 1

�1.rFi .Ox.t; �//; g.Ox.t; �/; t � t0//dt; (2.51)

where .; / denotes the inner product and � is a (generally) n-dimensional parameterentering in the homoclinic (or heteroclinic) solution Ox.t; �//. Now, however, mattersare more complicated, as we no longer have the geometric visualization of theinvariant manifolds and the distance between them that was available in then D 2 case. Thus, we shall not proceed any further with the discussion ofMel’nikov’s analysis and refer the interested reader to the literature for more details[90, 287, 288, 346].

Exercises

Exercise 2.1. (a) Perform the variable transformation x D sin.�=2/ to the equa-tion of motion of the simple pendulum (2.7) and show that this ODE becomes

Rx D ��

!2 C E

2

�x C 2!2x3; (2.52)

where ! D pg=l . Using the theory of Jacobi elliptic functions [6, 107, 169],

prove that the solution of this ODE can be expressed as the Jacobi elliptic sinefunction

x.t/ D C sn.�.t � t0/; �/; �2.1 C �2/ D !2 C E

2; �2�2 D !2C 2; (2.53)

which corresponds to the initial conditions x.t0/ D 0, Px.t0/ D �C , while theenergy equation gives 2�2C 2 D E . Eliminating C , � from these equations,obtain an expression relating the modulus � of the elliptic sine function to thetotal energy E and show that in the limit E ! 0, � ! 0, and x.t/ becomesthe usual trigonometric sine function. Prove also that in the limit � ! 1 thesolution on the separatrix of Fig. 2.4 reduces to the hyperbolic tangent function.

(b) Show that in the oscillatory domain the period T of the pendulum is given interms of the elliptic integral of the first kind

K.�2/ DZ �=2

0

d q

1 � �2 sin2

; (2.54)


and use this result to evaluate the first 2–3 terms in an expansion of T in powersof �2. Finally, use the Fourier representation of the elliptic sine function in termsof trigonometric sines to plot sn.u; �2/ as a function of its argument u for severalvalues of the modulus �2 (see [6, 107, 169]).

Exercise 2.2. Consider a system of two coupled harmonic oscillators of the typeconsidered in Sect. 2.2, Fig. 2.5. Call k D k1 the constant of the springs by whichthe two masses are tied to the walls and k D k2 the constant of the spring thatconnects them to each other. Find values k1; k2 such that the general solution of thesystem is periodic and determine the period in every case.

Exercise 2.3. Show that the unforced quadratic and cubic Duffing oscillators(2.40), (2.44) (for ı D ˙1) possess the Painleve property and solve the twoproblems completely in terms of Jacobi elliptic functions. Find explicit expressionsfor the solutions in all parts of the phase plane corresponding to regions of boundedand unbounded motion, as well as on the separatrices between these regions.

Exercise 2.4. Consider a Duffing cubic oscillator on which periodic forcing isapplied parametrically as follows

Pq D p; Pp D �q.1 C sin.˝t// C q3: (2.55)

This is a case where the spring constant (or the length of an associated pendulummodel) is taken to vary sinusoidally about its constant value. Note that the ellipticpoint at the origin and the two saddle points at .˙1; 0/ remain at their place when ¤ 0.

(a) Linearize (2.55) about the points .˙1; 0/ and obtain the corresponding linearstable and unstable manifolds Es , Eu . Keeping D 0 locate points onthese manifolds very close to the fixed points, which if propagated forward(or backward) in time by (2.55) will accurately trace the two separatrices S˙representing the upper and lower heteroclinic orbits joining the two saddles.

(b) When ¤ 0, construct the associated 2 � 2 linearized Poincare map L (and itsinverse L�1) by solving numerically (2.55) forward and backward in time andplotting points very close to .˙1; 0/ on the PSS ˙t0 at t D ˙2�=˝ . Evaluatethe linear stable and unstable manifolds Es , Eu of the 2 � 2 matrices L andL�1.

(c) Place many points on these linear manifolds very close to .˙1; 0/ and propagatethem forward (and backward) in time by solving numerically (2.55) andplotting their iterates at t D 2m�=˝ , m D ˙1; ˙2; : : :. Thus show thatthe corresponding nonlinear manifolds no longer join smoothly but intersecteach other repeatedly forming heteroclinic tangles, within which dense sets ofchaotic orbits can be found according to Smale horseshoe dynamics.


Problems

Problem 2.1. Perform all the calculations described in Sect. 2.2 to derive and studythe general solution of the Henon-Heiles problem in the integrable case A D B ,C D �1. Show that the two integrals in (2.30) are independent and in involution.Plot the invariant curves in the .X; PX / and .Y; PY / surfaces of section and relatethem to the exact solutions of the problem given in terms of Jacobi elliptic functions(see Exercise 2.1). Discuss the dynamics of the system for all values of the energyE > 0.

Problem 2.2. Complete all steps of the singularity analysis in the complex t-planeand show that the only cases of the Henon-Heiles Hamiltonian system possessingthe Painleve property are the ones listed in (2.34).

Problem 2.3. Use Mel’nikov’s theory as described in [160, 346] to derive anexpression for the distance between stable and unstable manifolds as a functionof the Mel’nikov integral and the magnitude of the unperturbed vector field alongthe invariant manifolds. Apply this theory to the case of the cubic Duffing oscillator(2.44) as outlined in Sect. 2.3.2, evaluate the corresponding Melnikov integral andprove the result shown in (2.49). Repeat this analysis for the parametrically drivenDuffing oscillator (2.55) and compare the results.

Problem 2.4. Study the cubic Duffing oscillator (2.43) in the case ı D �ˇ D�1, which describes a physical problem with a double well potential. Assumefurthermore that the periodic perturbation is of the form

Pq D p; Pp D q � q3 C ".��p C sin ˝t/; (2.56)

where ��p (with � > 0) is a term introducing dissipation. Draw the phase planecurves of the " D 0 case and show that the origin is a saddle fixed point whosestable and unstable manifolds are joined by two symmetric homoclinic solutions“embracing” two elliptic points at .0; ˙1/. Apply Mel’nikov’s theory to show thatfor " ¤ 0 these manifolds split into homoclinic orbits when � and satisfy acertain inequality. Now solve (2.56) numerically, using the approach described inExercise 2.4, to examine the validity of the inequality you derived for 0 < " � 1.How accurate are its predictions for the appearance of horseshoe chaos in theproblem?

Chapter 3Local and Global Stability of Motion

Abstract In this chapter, we discuss in a unified way equilibrium points, periodicorbits and their stability, which constitute local concepts of Hamiltonian dynamicstogether with ordered and chaotic motion, which are the concern of a more globaltype of analysis. Using the Fermi Pasta Ulam ˇ (FPU�ˇ) model as an example, westudy the destabilization properties of some of its simple periodic orbits and connectthem to wider aspects of the motion in phase space. We show numerically that thespectrum of positive Lyapunov exponents, characterizing a region of strong chaos,becomes invariant in the thermodynamic limit, verifying that the Kolmogorov-Sinai entropy, defined as the sum of these exponents, is an extensive property ofthe system. The chapter ends with an introduction of the smaller alignment index(SALI) and the generalized alignment index (GALI) criteria for distinguishingordered from chaotic motion.

3.1 Equilibrium Points, Periodic Orbits and Local Stability

3.1.1 Equilibrium Points

Let us note first that Hamilton’s equations of motion (1.18) can be written morecompactly in the form

Px D dxdt

D�

0N IN

�IN 0N

�rH.x/ D ˝rH.x/; x D .q; p/; (3.1)

where IN and 0N denote the N � N identity and zero matrices respectively. This isnot just a convenient notation. It introduces, in fact, the important matrix ˝ , whichis fundamental in establishing the symplectic structure of Hamiltonian dynamics.First of all, it enjoys a number of important properties:

.i/ ˝T D �˝ .antisymmetry/; .ii/ ˝T D ˝�1 .orthogonality/; .iii/ ˝�1 D �˝;

(3.2)


41

42 3 Local and Global Stability of Motion

based on which one defines the group of symplectic n � n matrices M as those thatsatisfy the condition

M D M˝M T ) M T D �˝M �1˝; M �1 D ˝M T˝T; (3.3)

with superscripts T and �1 denoting the transpose and inverse of a matrixrespectively. Using the definition of ˝ in (3.1), and its properties listed in (3.2)and (3.3) it is easy to show that det ˝ D 1 and det M D ˙1. In fact, it can beproved that the determinant of a symplectic matrix is exactly 1, i.e. det M D 1 (seeExercise 3.1).

As we discussed in Chap. 1, the simplest solutions of a Hamiltonian system areits equilibrium (or fixed points) ( Nq; Np), at which the right hand sides of Hamilton’sequations vanish,

@H. Nq; Np/

@pk

D 0;@H. Nq; Np/

@qk

D 0; k D 1; 2; : : : N: (3.4)

Thus, given a Hamiltonian system our first task is to find all its fixed points solvingthe nonlinear equations (3.4). Next, we need to examine the dynamics near each oneof these points.

To do this we write the solutions of (3.1) as small deviations about one of thefixed points as

x.t/ D . Nq; Np/ C �.t/; jj�.t/jj � "; (3.5)

(where " is the maximum of the Euclidean norms jj Nqjj; jj Npjj), substitute in (3.1) andlinearize Hamilton’s equations about this point to obtain

P�.t/ D A. Nq; Np/�.t/; (3.6)

where higher order terms in � have been omitted due to the smallness of the normof �.t/ noted in (3.5). The constant matrix A represents the Jacobian of ˝rH.x/

evaluated at the fixed point, as explained in (2.10) and (2.11). The importantobservation here is that A can be written as the product A D ˝S , where S D ST isa symmetric matrix. Thus, it can be easily shown that M D exp.A/ is symplecticand A is called a Hamiltonian matrix.

Clearly, the solutions of the linear system (3.6) will determine the local stabilitycharacter of . Nq; Np/ by telling us what kind of dynamics occurs in the vicinity of thisequilibrium point. As explained in Chap. 1, we shall call this fixed point linearlystable if all the solutions of (3.6) are bounded for all t . To find out under whatconditions this is true, let us write the general solution of this system as

�.t/ D eAt �.0/ D X.t/�.0/; (3.7)

where X.t/ is called the fundamental matrix of solutions of (3.6), with X.0/ D I2N ,the identity matrix. Clearly, the boundedness properties of these solutions depend

3.1 Equilibrium Points, Periodic Orbits and Local Stability 43

on the eigenvalues of the Hamiltonian matrix A. What do we know about theseeigenvalues? Many things, it turns out.

Observe that starting from the characteristic equation they satisfy det.A � �I/

D 0 and using the properties of symmetric matrices and ˝

det.A � �I/ D det.˝S � �I/ D det.˝S � �I/T D det.ST˝T � �I/ Ddet.�S˝ � �I/ D .�1/2N det.S˝ C �I/ D detŒ˝.S˝ C �I/˝�1� D

det.˝S C �I/ D det.A C �I/ D 0; (3.8)

we discover that if � is an eigenvalue of A so is ��. This implies that Hamiltoniansystems can never have asymptotically stable (or unstable) fixed points. We,therefore, conclude that a necessary and sufficient condition for such a equilibriumpoint to be stable is that all the eigenvalues of its corresponding matrix A have zeroreal part! And since A is a real matrix, if it possesses an eigenvalue � D ˛ C iˇ(with ˛, ˇ real), it will also have among its eigenvalues: � D ˛ � iˇ, � D �˛ C iˇ� D �˛ � iˇ.

This is great! Now we understand why in all the N D 1, N D 2 dof Hamiltoniansystems studied in Chap. 2, all stable fixed points have A matrices with purelyimaginary eigenvalues and the solutions in their neighborhood execute simpleharmonic motion. Furthermore, we also realize that a stable fixed point of aHamiltonian system can become unstable by two kinds of bifurcations:

1. A pair of imaginary .˙iˇ/ eigenvalues splitting on the real axis into an .˛; �˛/

pair, or2. An eigenvalue pair .˙iˇ/ splitting into four eigenvalues .˙˛ ˙ iˇ/ in the

complex plane, in a type of complex instability.

Bifurcation (1) leads to an equilibrium of the saddle type, since .˛; �˛/

correspond to two real eigenvectors, along which the solutions of (3.6) convergeor diverge exponentially from the fixed point. These eigenvectors identify theso-called stable and unstable euclidean manifolds, respectively, Es , Eu, of the fixedpoint. When continued under the action of the full nonlinear equations of motion(3.1) these become the stable and unstable invariant manifolds W s , W u, which mayintersect each other (or invariant manifolds of other saddle fixed points) and causethe horseshoe type of homoclinic (or heteroclinic) chaos we already encountered inthe examples of Chap. 2.

By contrast, bifurcation (2) occurs more rarely because it requires that the fourimaginary eigenvalues of a stable equilibrium point, .˙iˇ1; ˙ˇ2/, be degenerate,i.e. ˇ1 D ˇ2. It also does not arise in Hamiltonian systems of N D 1 dof (why?).As we will see in all the examples of Hamiltonian lattices analyzed in this book,bifurcation (1) is a lot more common and will thus quite appear frequently in thepages that follow.

At this point it is of crucial importance to make the following remark: Regardingthe stable and unstable manifolds of a saddle point, there are important theorems,which establish that the Es and Eu are tangent to W s and W u at the fixed point and


have respectively the same number of dimensions [175, 257]. This fact is of greatpractical importance! It allows us to ‘trace out’ numerically with great accuracythe W s and W u, starting with many initial conditions distributed on the Es andEu very close to the fixed point and integrating (3.1) forward in time to obtainW u and backward to obtain W s . As we will find out in Chap. 7, this strategy willprove extremely useful when we need to construct discrete breather solutions usingthe (homoclinic or heteroclinic) intersections of W s and W u invariant manifolds ofinvertible maps.

3.1.2 Periodic Orbits

Enough said of equilibrium points. It is time now to discuss the next most importanttype of solution of Hamiltonian systems, which is their periodic orbits. You mightexpect, of course, the mathematics here to become more involved and you would beright. However, as we will soon find out, the wonderful instrument of the Poincaremap and its associated surfaces of section will come to the rescue and make theanalysis a lot easier. Let us begin by giving a more general definition of the Poincaremap than the one we used in Chap. 2.

In particular, we will assume that our n-dimensional dynamical system, cast inthe general form Px D f.x/ (see (1.1)) has a periodic solution Ox.t/ D Ox.t C T / ofperiod T . Let us choose an arbitrary point along this orbit Ox.t0/ and define a PSS atthat point as follows

˙t0 D fx.t/ = .x.t/ � Ox.t0// � f.Ox.t0// D 0g : (3.9)

Thus, ˙t0 is a .n � 1/-dimensional plane which intersects the given periodic orbit atOx.t0/ and is vertical to the direction of the flow at that point. Clearly now a Poincaremap can be defined on that plane as before, by

P W ˙t0 ! ˙t0; xkC1 D P xk; k D 0; 1; 2; : : : (3.10)

for which x0 D Ox.t0/ is a fixed point, since x0 D P x0. We now examine smalldeviations about this point,

xk D Ox0 C �k; jj�kjj � "; (3.11)

(where " is of the same magnitude as jjOx0jj), substitute (3.11) in (3.10) and linearizethe Poincare map to obtain

�kC1 D DP.Ox0/�k; (3.12)

3.1 Equilibrium Points, Periodic Orbits and Local Stability 45

where we have neglected higher order terms in � and DP.Ox0/ denotes the Jacobianof P evaluated at Ox0.

“But we don’t know anything about P !”, you will rightfully argue. Don’t worry!We always have the variational equations of the original differential equationsderived by writing x.t/ D Ox.t/ C �.t/, whence linearizing (1.1) about this periodicorbit leads to the system

P�.t/ D A.t/�.t/; A.t/ D A.t C T /; (3.13)

where A.t/ is the Jacobian matrix of f.x/ evaluated at the periodic orbit x.t/ D Ox.The crucial question, of course, we must face now is: How are the two linear systems(3.12) and (3.13) related to each other?

The careful reader will have certainly observed that we have used differentnotations for the small deviations about the periodic orbit: �.t/ in the continuoustime setting of differential equations and �k in the discrete time setting of thePoincare map. This is not just because they represent different quantities, it is alsoto emphasize that their dimensionality as vectors in the n-dimensional phase spaceR

n (n D 2N for a Hamiltonian system) is different: �.t/ is n-dimensional, while �k

is .n � 1/-dimensional! Now you may be feeling again somewhat pessimistic. Howare we ever to match these two small deviation variables?

The answer will come from what is called Floquet theory [107, 265, 346]. Firstwe realize that since (3.13) is a linear system of ODEs it must possess, in general,n linearly independent solutions, forming the columns of the n � n fundamentalsolution matrix M.t; t0/ in

�.t/ D M.t; t0/�.0/; M.t; t0/ D M.t C T; t0/ (3.14)

(see (3.7)). Now, we don’t know what these solutions are. If we were, however, tochange our basis at the point Ox.t0/ so that one of the directions of motion is alongthe direction vertical to the PSS (3.9), we would observe that the nth column ofthe matrix M.T; t0/ has zero elements except at the last entry which is 1. Thus,if we eliminate from this matrix its nth row and nth column, it turns out that its.n � 1/ � .n � 1/ submatrix is none other than our beloved Poincare map (3.10)!Surprised? That is the relation between the two approaches we were seeking.

“So what?” you may ask. How does that help us? Well, it means that if we couldcompute the so-called monodromy matrix M.T; t0/ numerically we could evaluateits eigenvalues, �1; : : : ; �n�1 (the last one being �n D 1), which are those of thePoincare map and determine the stability of our periodic orbit as follows: If they areall on the unit circle, i.e. j�i j D 1; i D 1; : : : ; n � 1, the periodic orbit is (linearly)stable, while if (at least) one of them satisfies j�j j > 1 the periodic solution isunstable (see Exercise 3.2).

But how do we compute the monodromy matrix M.T; t0/? It is not so difficult.Let us first set t0 D 0 for convenience and observe from (3.14) that M.0; 0/ D In.All we have to do now is integrate numerically the variational equations (3.14) fromt D 0 to t D T , n times, each time for a different initial vector .0; : : : ; 0; 1; 0; : : : ; 0/


with 1 placed in the i th position, i D 1; 2; : : : ; n. Note that since these equationsare linear numerical integration can be performed to arbitrary accuracy and is alsonot too-time consuming for reasonable values of the period T .

Once we have calculated M.0; 0/, we may proceed to compute its eigenvaluesand determine the stability of the periodic orbit according to whether at least one ofthese eigenvalues has magnitude greater than 1. In fact, for Hamiltonian systems,it can be shown that M.0; 0/ is a 2N � 2N symplectic matrix and hence theproduct of its eigenvalues is equal to 1 (see Exercise 3.1). Furthermore, symplecticmatrices possess the very important property that half of their eigenvalues arethe inverse of the remaining half, i.e. their full eigenvalue spectrum has the form�1; : : : ; �n; 1=�1; : : : ; 1=�n (see Exercise 3.1).

These are very powerful results. First they provide us with a precise and easilyimplementable strategy for determining the stability of any periodic orbit of aHamiltonian system of N dof. They also demonstrate that for such a periodic orbitto be (linearly) stable, all eigenvalues of its monodromy matrix must in general becomplex and lie on the unit circle. For, if one of them is real (and different from 1),there will be an eigenvalue greater than 1 and nearby displacements from the orbitwill grow exponentially at least in one direction in the 2N D phase space R2N .

In Problems 3.1 and 3.2 at the end of the chapter we urge the reader to apply theabove theory to a periodically driven cubic Duffing oscillator and a special case ofthe N D 2 dof Henon-Heiles Hamiltonian. Only if you solve these problems youwill be able to appreciate the more elaborate applications of Floquet theory in thesections that follow. More importantly, however, these problems will also teach youthat stability of periodic orbits once relinquished is not lost forever! It may indeedbe recovered (even infinitely) many times as we vary an important parameter likethe total energy of the system E .

3.2 Linear Stability Analysis

Now that we have learned how to study the linear stability properties of periodicsolutions of Hamiltonian systems, it is time to wonder about the implications ofthis analysis regarding the more “global” dynamics, which is really what we areinterested in. Let us turn, therefore, immediately to the class of one-dimensionallattices (or chains) of coupled oscillators, to which this book is primarily devoted.

Our first example is the famous Fermi Pasta Ulam (FPU)�ˇ model described bythe N dof Hamiltonian [43]

H D 1

2

NXj D1

p2j C

NXj D0

1

2.xj C1 � xj /2 C 1

4ˇ.xj C1 � xj /4 D E; (3.15)

where xj are the displacements of the particles from their equilibrium positions,and pj D Pxj are the corresponding canonically conjugate momenta, ˇ is a positive

3.2 Linear Stability Analysis 47

real constant and E is the total energy of the system. Note that by not including anycubic nearest neighbor interactions in (3.15), we have kept an important symmetryof the system under the interchange xj ! �xj , which will make our analysissimpler. However, most of what we shall be discussing can also be studied whencubic interactions are included with an ˛ coefficient before them in what is calledthe FPU-˛ model (see Chaps. 6 and 7).

The second example we shall use as an illustration is called the BEC Hamiltonianand is obtained by a discretization of a PDE of the nonlinear Schrodinger type calledthe Gross-Pitaevskii equation [106],

ih@�.x; t/

@tD �h2

2

@2�.x; t/

@x2C V.x/�.x; t/ C gj�.x; t/j2�.x; t/; i2 D �1

(3.16)where h is Planck’s constant, g is a positive constant (repulsive interactions betweenatoms in the condensate) and V.x/ is an external potential. This equation is relatedto the phenomenon of Bose-Einstein Condensation (BEC) [187]. Considering thesimple case V.x/ D 0, h D 1 and discretizing the complex variable �.x; t/ ��j .t/, we may approximate the second order derivative by

�xx ' �j C1 C �j �1 � 2�j

ıx2; �j .t/ D qj .t/ C ipj .t/; j D 1; 2; : : : ; N; (3.17)

and by definingj�.x; t/j2 D q2

j .t/ C p2j .t/; (3.18)

obtain a set of ODEs for the canonically conjugate variables, pj and qj , describedby the BEC Hamiltonian [318, 331]

H D 1

2

NXj D1

.p2j Cq2

j /C �

8

NXj D1

.p2j Cq2

j /2 � "

2

NXj D1

.pj pj C1 Cqj qj C1/ D E; (3.19)

where � > 0 and " D 1 are constant parameters, g D �=2 > 0, with ıx D 1 andE the total energy. In fact, besides the energy integral the BEC Hamiltonian (3.19)possesses one more constant of the motion given by the norm quantity

F DNX

j D1

.p2j C q2

j / (3.20)

and therefore chaotic behavior can only occur for N � 3.Let us focus on a special class of periodic solutions we have called Simple

Periodic Orbits (SPOs), which have well-defined symmetries and are known inclosed form. By SPOs, we mean periodic solutions where all variables return totheir initial state after only one maximum and one minimum in their oscillation. Inparticular the SPOs we shall be concerned with are the following:


I. For the FPU and BEC with periodic boundary conditions:

xN Ck.t/ D xk.t/; 8t; k: (3.21)

(a) The Out-of-Phase Mode (OPM) for FPU, with N even (often called the �-mode)

Oxj .t/ D � Oxj C1.t/ � Ox.t/; j D 1; : : : ; N: (3.22)

(b) The OPM for BEC, with N even

qj .t/ D �qj C1.t/ D Oq.t/; j D 1; : : : ; N: (3.23)

(c) The In-Phase Mode (IPM) for BEC

qj .t/ D Oq.t/; pj .t/ D Op.t/; 8j D 1; : : : ; N; N 2 N and N � 2: (3.24)

II. For the FPU model and fixed boundary conditions:

x0.t/ D xN C1.t/ D 0; 8t: (3.25)

(a) The SPO1 mode, with N odd,

Ox2j .t/ D 0; Ox2j �1.t/ D � Ox2j C1.t/ � Ox.t/; j D 1; : : : ;N � 1

2: (3.26)

(b) The SPO2 mode, with N D 5 C 3m; m D 0; 1; 2; : : : particles,

x3j .t/ D 0; j D 1; 2; 3 : : : ;N � 2

3;

xj .t/ D �xj C1.t/ D Ox.t/; j D 1; 4; 7; : : : ; N � 1: (3.27)

In later chapters, we will see that the above SPOs are examples of what wecall Nonlinear Normal Modes (NNMs), whose theory is discussed in detailin Chap. 4.

To appreciate the importance of these periodic orbits, let us consider the N D 2

case of the BEC Hamiltonian, which is completely integrable. Plotting in Fig. 3.1its PSS .x; px/ (for y D 0, py > 0), we observe that the IPM and OPM orbits(intersecting the vertical axis at the points P1, P2 respectively) play a major rolein the global dynamics: In Fig. 3.1a they are both stable with large size islands ofperiodic and quasiperiodic tori around them. On the other hand, the local picturechanges dramatically when one of them becomes unstable, e.g. when we increasethe value of the norm integral (3.20), as in Fig. 3.1b. Of course, the N D 2 dofBEC Hamiltonian is integrable and no chaos arises due to the destabilization of theOPM orbit. One may very well wonder, however, what would happen after a similarbifurcation of the OPM orbit in the BEC system with N � 3 dof? How widespreadwould chaos be around the unstable orbit in that case?


ba

Fig. 3.1 PSS of the BEC Hamiltonian, showing the IPM and OPM orbits intersecting the verticalaxis at the points P1 and P2 respectively. (a) The value of the norm integral is F D 2, whilethe SPOs correspond to different values of the Hamiltonian. (b) The value of the norm integral isF D 4:1, where the OPM on the negative px axis has become unstable yielding two stable SPOsone above and one below it (after [52])

Fortunately, the FPU system with fixed boundary conditions is one of thoseexamples where we can directly apply Lyapunov’s Theorem 1.3 of Chap. 1: Morespecifically, we can use it to prove the existence of SPOs as continuations of thelinear normal modes of the system, whose frequencies have the well-known form[43, 88, 128, 133, 226]

!q D 2 sin

��q

2.N C 1/

�; q D 1; 2; : : : ; N: (3.28)

This is so because the linear mode frequencies (3.28) are seen to satisfy Lyapunov’snon-resonance condition for the !qs, stated in Theorem 1.3 for all q and generalvalues of N . Thus, our SPO1 and SPO2 orbits, as NNMs of the FPU Hamiltonian,are identified by the indices q D .N C 1/=2 and q D 2.N C 1/=3 respectively.

As we discussed in the previous section, linear stability analysis of periodicsolutions can be performed by studying the eigenvalues of the monodromy matrix.For example, consider the SPO1 mode of FPU: The equations of motion

Rxj .t/ D xj C1 � 2xj C xj �1 C ˇ�.xj C1 � xj /3 � .xj � xj �1/

3�; j D 1; : : : ; N;

(3.29)for fixed boundary conditions collapse to a single second order ODE:

ROx.t/ D �2 Ox.t/ � 2ˇ Ox3.t/: (3.30)

Its solution is well-known in terms of Jacobi elliptic functions

Ox.t/ D C cn.�t; �2/; (3.31)


where

C 2 D 2�2

ˇ.1 � 2�2/; �2 D 2

1 � 2�2; (3.32)

and �2 is the modulus of the cn elliptic function.Its stability is analyzed setting xj D Oxj C yj and keeping up to linear terms in

yj to obtain the variational equations

Ryj D .1 C 3ˇ Ox2/.yj �1 � 2yj C yj C1/; j D 1; : : : ; N; (3.33)

where y0 D yN C1 D 0. We now separate these variational equations into N

uncoupled Lame equations

Rzj .t/ C 4.1 C 3ˇ Ox2/sin2

��j

2.N C 1/

�zj .t/ D 0; j D 1; : : : ; N; (3.34)

where the zj variations are simple linear combinations of the yj ’s (see Exercise 3.3).Changing variables to u D �t , this equation takes the form

z00j .u/ C 2

�1 C 4�2 � 6�2sn2.u; �2/

�sin2

��j

2.N C 1/

�zj .u/ D 0; j D 1; : : : ; N;

(3.35)where primes denote differentiation with respect to u. Equation 3.35 is an exampleof Hill’s equation [240, 250, 344]

z00.u/ C Q.u/z.u/ D 0; (3.36)

where Q.u/ is a T -periodic function, i.e. Q.u/ D Q.u C T / with T D 2K andK D K.�2/ is the elliptic integral of the first kind (see (2.54)). In Exercise 3.3 weask the reader to repeat the steps of the above analysis to analyze the stability of theother NNMs of the FPU and BEC systems mentioned above. In particular, we askyou to try to think of other such nonlinear modes, for which, due to symmetries, theN equations of motion reduce to a single nonlinear oscillator. If not, can you thinkof other NNMs (based on symmetry arguments), which reduce to the solution ofn D 2; 3; : : :, coupled oscillators? This is a topic we shall examine in detail whenwe discuss bushes of orbits in Chap. 4.

Thus, we now have two ways for studying the stability of these fundamentalperiodic orbits [13,16]: One is to compute numerically the fundamental solutions of(3.34) and determine whether the eigenvalues of the monodromy matrix M.T; 0/ lieon the unit circle or not. In fact, we don’t need to do this for all N of these equations.It suffices to study only the variation zm.u/ (in the SPO1 case it is m D .N � 1/=2),which is the first to become unbounded as �2 (or the energy E) increases.

The other method is to apply the theory of Hill’s equation [240, 250, 344] tothe crucial j D m Lame equation (3.35) and determine the stability of the NNMaccording to the condition


a b

Fig. 3.2 (a) The solid curve corresponds to the energy per particle Ec=N , for ˇ D 1, of thefirst destabilization of the SPO1 nonlinear mode of the FPU system (3.15) with fixed boundaryconditions, obtained by the numerical evaluation of the monodromy matrix, while the dashed linecorresponds to the function / 1=N . (b) Same as in (a) with the solid curve depicting the firstdestabilization of the OPM periodic solution of the FPU system (3.15). Here, however, the dashedline corresponds to the function / 1=N 2. Note that both axes are logarithmic (after [52])

ˇˇ1 � 2sin2

�K.�2/

pa0

�det .D.0//

ˇˇ D

�< 1; stable mode> 1; unstable mode

; (3.37)

where D.0/ is an infinite Hill’s matrix, whose entries are given in terms of theFourier coefficients of the NNM and whose value is well approximated by rapidlyconvergent n� n approximants [13,16]. Both ways lead to the interesting result thatthe critical energy threshold values for the first destabilization of the SPO1 solutionsatisfies Ec=N / 1=N , while for the OPM solution (3.22) we find Ec=N / 1=N 2,as shown in Fig. 3.2a, b respectively.

What does all this mean? Let us try to find out.

3.2.1 An Analytical Criterion for “Weak” Chaos

As we discovered from the above analysis, the NNMs of the FPU and BECHamiltonians studied so far experience a first destabilization at energy densitiesof the form

Ec

N/ N �˛; ˛ D 1; or; 2; N ! 1: (3.38)

This means that for fixed N , some of these fundamental periodic solutions becomeunstable at much lower energy than others. We may, therefore, expect that those thatdestabilize at lower energies will have smaller chaotic regions around them, as thegreater part of the constant energy surface is still occupied by tori of quasiperiodicmotion. Could we then perhaps argue that near NNMs characterized by the exponent˛ D 2 in (3.38) one would find a “weaker” form of chaos than in the ˛ D 1 case?


This indeed appears to be true at least for the FPU Hamiltonian model. As wasrecently shown in [128], the energy threshold for the destabilization of the lowq D 1; 2; 3; : : :, nonlinear modes, representing continuations of the correspondinglinear model (see (3.28)), satisfies the analytical formula

Ec

N� �2

6ˇN.N C 1/; (3.39)

and is therefore of the type ˛ D 2 in (3.38). Remarkably enough this local loss ofstability coincides with the “weak” chaos threshold shown in [109, 110] to haveglobal consequences regarding the dynamics of the system as a whole, as it isassociated with the breakup of the famous FPU recurrences! In Chap. 6 we shallexamine this very important phenomenon in detail. For the moment, let us simplypoint out that the destabilization of individual NNMs occurring at low energiesappears to be somehow related to a transition from a “weak” to a “stronger” type ofchaos in the full N particle chain.

Interestingly enough, it was later discovered [13, 16] that the energy threshold(3.39) for the low q modes, also coincides with the instability threshold of ourSPO2 mode which corresponds to q D 2.N C 1/=3! In Fig. 3.3 we comparethe approximate formula (dashed line) with our destabilization threshold for SPO2obtained by the monodromy matrix analysis (solid line) for ˇ D 0:0315 and findexcellent agreement especially in the large N limit.

We may, therefore, arrive at the following conclusions based on the above results:Linear stability (or instability) of periodic solutions is certainly a local property andcan only be expected to reveal how orbits behave in a limited region of phase space.And yet, we find that if these periodic solutions belong to the class of nonlinearcontinuations of linear normal modes, their stability character may have importantconsequences for the global dynamics of the Hamiltonian system. In particular, ifthe exponent of their first destabilization threshold in (3.38) is ˛ D 2 they areconnected with the onset of “weak” chaos as a result of the breakdown of FPU

Fig. 3.3 The solid curvecorresponds to the energyE2u.N / of the firstdestabilization of the SPO2mode of the FPU system(3.15) with fixed boundaryconditions and ˇ D 0:0315

obtained by the numericalevaluation of the eigenvaluesof the monodromy matrix.The dashed line correspondsto the approximate formula(3.38) for the q D 3 nonlinearnormal solution (after [13])

3.3 Lyapunov Characteristic Exponents and “Strong” Chaos 53

recurrences. On the other hand, if ˛ D 1 as in the case of the SPO1 mode, they arisein much wider chaotic domains and have orbits in their neighborhood which evolvefrom “weak” to “strong” chaos passing through quasi-periodic states of varyingcomplexity, with different statistical properties, as we describe in detail in Chap. 8.

3.3 Lyapunov Characteristic Exponents and “Strong” Chaos

3.3.1 Lyapunov Spectra and Their Convergence

Let us now study the chaotic behavior in the neighborhood of our unstable SPOs,starting with the well-known method of the evaluation of the spectrum of Lyapunovcharacteristic exponents (LCEs) of a Hamiltonian dynamical system,

Li ; i D 1; : : : ; 2N; L1 � Lmax > L2 > : : : > L2N : (3.40)

The LCEs measure the rate of exponential divergence of initially nearby orbitsin the phase space of the dynamical system as time approaches infinity. InHamiltonian systems, the LCEs come in pairs of opposite sign, so their sumvanishes,

P2NiD1 Li D 0, and two of them are always equal to zero corresponding

to deviations along the orbit under consideration. If at least one of them (the largestone) L1 � Lmax > 0, the orbit is chaotic, i.e. almost all nearby orbits divergeexponentially in time, while if Lmax D 0 the orbit is stable (linear divergence ofinitially nearby orbits). The numerical algorithm we use here for the computation ofall LCEs is the one proposed in [31, 32]. A detailed review of the theory of LCEsand the numerical techniques used for their evaluation can be found in [310].

In theory, Li � Li .x.t// for a given orbit x.t/ expresses the limit for t ! 1 ofa quantity of the form

Kit D 1

tln

k wi .t/ kk wi .0/ k ; (3.41)

Li D limt!1 Ki

t (3.42)

where wi .0/ and wi .t/; i D 1; : : : ; 2N � 1 are infinitesimal deviation vectors fromthe given orbit x.t/ (at times t D 0 and t > 0 respectively) that are orthogonal to thevector tangent to the orbit (since the LCE in the direction along the orbit is zero).The time evolution of wi is given by solving the variational equations of the system,i.e. the linearized equations about the orbit, assuming that the limit of (3.42) existsand converges to the same Li , for almost all choices of initial deviations wi .0/.

In practice, however, the above computation is more involved [31, 32]: Sincethe exponential growth of wi .t/ is observed for short time intervals, one stops theevolution of wi .t/ after some time T1, records the computed Ki

T1, orthonormalizes


a b

c d

Fig. 3.4 (a) Lyapunov spectra of neighboring orbits of the SPO1 (solid lines) and SPO2 (dashedlines) modes of the FPU model (3.15) with fixed boundary conditions, for N D 11, at (a) energiesE D 1:94 (SPO1) and E D 0:155 (SPO2), where both modes have just destabilized, (b) energyE D 2:1 for both SPOs, where the spectra are still distinct, (c) and convergence of the Lyapunovspectra of neighboring orbits of the SPOs at energy E D 2:62 where both of them are unstable.(d) Coincidence of Lyapunov spectra continues at energy E D 5 (after [13])

the vectors wi .t/ and repeats the calculation for the next time interval T2 (witht D 0 replaced by t D T1), etc. obtaining finally Li as an average over many suchTj , j D 1; 2; : : : ; n as

Li D 1

n

nXj D1

KiTj

; n ! 1: (3.43)

For fixed N it has been found that as E increases, the Lyapunov spectrum (3.40) forall the unstable NNMs studied in the previous section appears to fall on a smoothcurve [13, 16].

Let us examine this in more detail by plotting the Lyapunov spectra of twoneighboring orbits of the SPO1 and SPO2 modes, of the FPU system with fixedboundary conditions, for N D 11 dof and energy values E1 D 1:94 and E2 D 0:155

respectively, where both SPOs have just destabilized (see Fig. 3.4a). Here, themaximum Lyapunov exponents L1, are very small .�10�4/ and the correspondingLyapunov spectra are quite distinct.

Raising now the energy to E D 2:1, we observe in Fig. 3.4b that the Lyapunovspectra of the two SPOs are closer to each other, but still quite different. AtE D 2:62, however, we see that the two spectra have nearly converged to the sameexponentially decreasing function

3.3 Lyapunov Characteristic Exponents and “Strong” Chaos 55

Li .N / / e�˛ iN ; i D 1; 2; : : : ; N; (3.44)

and their maximal Lyapunov exponents are virtually the same. The ˛ exponentsfor the SPO1 and SPO2 are found to be approximately 2:3 and 2:32 respectively.Figure 3.4d also shows that this coincidence of Lyapunov spectra persists at higherenergies.

3.3.1.1 Lyapunov Spectra and the Thermodynamic Limit

Let us now choose our initial conditions near unstable SPOs of our Hamiltoniansand try to determine some important statistical properties of the dynamics in the so-called thermodynamic limit, where we let E and N grow indefinitely keeping E=N

constant. In particular, we will compute the spectrum of the Lyapunov exponentsof the FPU and BEC systems starting near the OPM solutions (3.22) and (3.23) forenergies where these orbits are unstable. We thus find that the Lyapunov spectra arewell approximated by smooth curves of the form Li � L1e�˛i=N , for both systems,with ˛ � 2:76, ˛ � 3:33 respectively (see Fig. 3.5).

The above exponential formula is found to hold quite well, up to i D K.N / �3N=4. For the remaining exponents, the spectrum is seen to obey different decaylaws, which are not easy to determine. In fact, such Lyapunov spectra provideimportant invariants of the dynamics, in the sense that, in the thermodynamic limit,we can use them to evaluate the average of the positive LCEs, yielding the so-calledKolmogorov-Sinai entropy hKS .N / [165, 267] for each system and find that it is aconstant characterized by the value of the maximum Lyapunov exponent (MLE) L1

and the exponent ˛ appearing in them.Thus, we plot in Fig. 3.6 the Kolmogorov-Sinai entropy (solid curves), which is

defined as the sum of the N � 1 positive Lyapunov exponents,

hKS.N / DN �1XiD1

Li .N /; Li .N / > 0: (3.45)

a b

Fig. 3.5 The spectrum of positive LCEs (3.40) of (a) the OPM (3.22) of the FPU Hamiltonian(3.15) for fixed E=N D 3=4, (b) the OPM (3.23) of the BEC Hamiltonian (3.19) for fixed E=N D3=2. For both models we consider periodic boundary conditions and i runs from 1 to N (after [16])


a b

Fig. 3.6 The hKS .N / entropy near the OPM of (a) the FPU Hamiltonian for fixed E=N D 3=4

(solid curve) and the approximate formula (dashed curve), and (b) of the BEC Hamiltonian forfixed E=N D 3=2 (solid curve) and the approximate formula (dashed curve) (after [16])

In this way, we find, for both Hamiltonians, that hKS.N / is an extensive thermo-dynamic quantity as it is clearly seen to grow linearly with N (hKS.N / / N ),demonstrating that in their chaotic regions the FPU and BEC Hamiltonians behaveas ergodic systems of BG statistical mechanics.

Finally, assuming that the exponential law Li � L1e�˛i=N is valid for all i D1; : : : ; N � 1, we approximate the sum of the positive Lyapunov exponents Li , andcalculate the hKS.N / entropy from (3.45) as

hKS.N / / Lmax1

�6 C e2:76N

; Lmax � 0:099; (3.46)

for the OPM of the FPU model and

hKS.N / / Lmax1

�1 C e3:33N

; Lmax � 0:34; (3.47)

for the OPM of the BEC Hamiltonian. In Fig. 3.6, we have plotted (3.46) and (3.47)with dashed curves and obtain nearly straight lines with the same slope as the datacomputed by the numerical evaluation of the hKS .N /, from (3.45). The reader isalso advised to see [229] where analogous results are obtained for the Lyapunovspectra in the thermodynamic limit and the computation of hKS.N / in the FPU-ˇmodel.

3.4 Distinguishing Order from Chaos

The reader must have realized by now that if we wish to make serious progress in thestudy of the dynamics of Hamiltonian systems we must be able to develop accurateand efficient tools for distinguishing between order and chaos, both locally and

3.4 Distinguishing Order from Chaos 57

globally. This means that these tools must be able to: (a) characterize correctly thetime evolution of any given set of initial conditions and (b) approximately identifyregimes of ordered vs. chaotic motion on a constant .2N � 1/-dimensional energysurface in the phase space R2N .

“But, surely”, you will argue, “we already possess such tools. They are theLyapunov exponents, whose usefulness was already evident in the results of theprevious section”. This is indeed true and we shall frequently mention theseexponents in the chapters that follow. We must note, however, that powerful as theymay be they also have a serious drawback: Their values vary significantly in timeand may only be used in the long time limit when the exponents have convergedwith satisfactory accuracy.

Furthermore, it is well-known that a positive MLE not only implies chaoticbehavior, it also quantifies it: As is commonly observed, the larger the MLE valuethe “stronger” the chaotic properties, i.e. the faster the divergence of nearby orbits.What happens, however, when we are very close to a region of ordered motion,where the MLE converges very slowly to a non-zero value, if it does so at all? Andhow do we know if it is not exactly zero and we are in fact following a quasiperiodicrather than a chaotic orbit?

It is for this reason that many researchers have developed over the last twodecades, alternative approaches to characterize orbits of Hamiltonian systems asordered or chaotic. These methods can be divided in two major categories, the oneswhich are based on the evolution of deviation vectors from a given orbit, like thecomputation of MLE, and the ones which rely on the analysis of the particular orbitunder study.

Among other chaoticity detectors belonging to the same category with theevaluation of the MLE, are the fast Lyapunov indicator (FLI) [135, 137, 138] andits variants [26, 27], the smaller alignment index (SALI) [309, 313, 314] and itsgeneralization, the so-called generalized alignment index (GALI) [247, 315, 316],the mean exponential growth of nearby orbits (MEGNO) [95, 96], the relativeLyapunov indicator (RLI) [296, 297], as well as methods based on the study ofspectra of quantities related to the deviation vectors like the stretching numbers[136, 231, 340], the helicity angles (the angles of deviation vectors with a fixeddirection) [102], the twist angles (the differences of two successive helicity angles)[103], or the study of the differences between such spectra [217, 341].

In the category of methods based on the analysis of a time series constructed bythe coordinates of the orbit under study, one may list the frequency map analysis ofLaskar [211–214], the ‘0-1’ test [153,154], the method of the low frequency spectralanalysis [204,342], the ‘patterns method’ [305], the recurrence plots technique [354,355] and the information entropy index [258]. One could also refer to several ideaspresented by various authors that could be used to distinguish order from chaos,like the differences appearing for ordered and chaotic orbits in the time evolutionof their correlation dimension [134], the time averages of kinetic energies relatedto the virial theorem [171] and the statistical properties of series of time intervalsbetween successive intersections of orbits with a PSS [181].


To our knowledge, a systematic and detailed comparative study of the efficiencyand reliability of the various chaos detection techniques is not presently available.Still, a number of comparisons between some of the existing methods have beenperformed on various examples of dynamical systems [28, 239, 309, 314, 324].

Of all these methods, we have chosen to dwell on two indicators that the authorsof this book have found most convenient and useful in their studies: the SALI andthe GALI. These indicators are derived from a detailed analysis of the variationalequations describing the tangent space of the orbits as time evolves. They areaccurate and efficient in the sense that they: (a) correctly identify the chaotic natureof the orbits more rapidly than other methods and, perhaps more importantly,(b) successfully characterize quasiperiodic motion providing also the number ofdimensions of the torus on which it lies. In the next subsections we discuss brieflythe SALI and GALI indicators and return to examine them more carefully inlater chapters, when we address more delicate issues regarding the complexity ofHamiltonian dynamics.

Before proceeding, however, it is instructive to point out a crucial differencebetween Hamiltonian (more generally conservative) and dissipative systems. Asthe reader has realized by now, chaotic regions and tori of ordered motion inHamiltonian (and also conservative) systems are distributed in very complicatedways and are present down to infinitesimally small scales. Thus, the indicatorsmentioned above find their greatest usefulness in the conservative case, where thedistinction between order and chaos is often a very delicate issue. When dissipationis added, the detailed features of the dynamics are smoothed out and all orbitsconverge either to simple (fixed points, periodic orbits, isolated tori) or strangeattractors [160, 346]. In such cases, the identification of chaotic vs. ordered motionbecomes a simple matter, which is usually settled by simply computing Lyapunovexponents and in particular the MLE.

3.4.1 The SALI Method

The SALI method was originally developed to distinguish between ordered andchaotic orbits in symplectic maps (see Sect. 5.1 for the definition of a symplecticmap) and Hamiltonian systems [309, 313, 314], and was soon applied by manyresearchers to a number of important examples [55, 57, 72, 233, 234, 242, 245, 262,269, 319, 320, 322, 325]. To compute this indicator, one follows simultaneously thetime evolution of a reference orbit along with two deviation vectors with initialconditions w1.0/, w2.0/, normalizing them from time to time to 1, as follows

Owi .t/ D wi .t/

kwi .t/k ; i D 1; 2: (3.48)

The SALI is then defined as

SALI.t/ D min fk Ow1.t/ C Ow2.t/k ; k Ow1.t/ � Ow2.t/kg : (3.49)


In references [309, 314] it has been shown that, in the case of chaotic orbits, thedeviation vectors Ow1, Ow2 eventually become aligned in the direction of the MLE andSALI.t/ falls exponentially to zero as

SALI.t/ / e�.L1�L2/t ; (3.50)

L1, L2 being the two largest LCEs. In the case of ordered motion, on the other hand,the orbit lies on a torus and eventually the vectors Ow1, Ow2 fall on the tangent space ofthe torus, following a t�1 time dependence. In this case, the SALI oscillates aboutvalues that are different from zero [309, 313], i.e.

SALI � const: > 0; t ! 1: (3.51)

Thus, the different behavior of the index for ordered and chaotic orbits allows usto clearly distinguish between the two cases.

3.4.2 The GALI Method

The GALI is an efficient chaos detection technique introduced in [315] as ageneralization of SALI. This generalization consists in the fact that GALI usesinformation of more than two deviation vectors from the reference orbit, leadingto a faster and clearer distinction between regular and chaotic motion than SALI.The method has been applied successfully to different dynamical systems for thediscrimination between order and chaos, as well as for the detection of quasiperiodicmotion on low dimensional tori [14, 63, 93, 242–244, 246, 316].

The Generalized Alignment Index of order k (GALIk), 2 k 2N , isdetermined through the evolution of k initially linearly independent deviationvectors wi .0/. As in the case of SALI, deviation vectors wi .t/ are normalized fromtime to time to avoid overflow problems, but their directions are left intact. Thus,according to [315] GALIk is defined as the volume of the k-parallelepiped havingas edges the k unitary deviation vectors Owi .t/ D wi .t/=kwi .t/k, i D 1; 2; : : : ; k,determined through the wedge product of these vectors as

GALIk.t/ D k Ow1.t/ ^ Ow2.t/ ^ � � � ^ Owk.t/k; (3.52)

where k � k denotes the usual norm. From this definition it is evident that if at leasttwo of the deviation vectors become linearly dependent, the wedge product in (3.52)becomes zero and the GALIk vanishes.

In the case of a chaotic orbit, all deviation vectors tend to become linearlydependent, aligning in the direction defined by the MLE, and GALIk tends to zeroexponentially following the law [315]

GALIk.t/ / e�Œ.L1�L2/C.L1�L3/C��C.L1�Lk/�t ; (3.53)

where L1; : : : ; Lk are the k largest LCEs.


In the R2N phase space of an N dof Hamiltonian flow or a 2N D map, regular

orbits lie on s-dimensional tori, with 2 s N for Hamiltonian flows, and1 s N for maps. For such orbits, all deviation vectors tend to fall on thes-dimensional tangent space of the torus on which the motion lies. Thus, if westart with k s general deviation vectors they will remain linearly independenton the s-dimensional tangent space of the torus, since there is no particular reasonfor them to become aligned. As a consequence, GALIk remains practically constantand different from zero for k s. On the other hand, GALIk tends to zero fork > s, since some deviation vectors will eventually become linearly dependent.In particular, the generic behavior of GALIk for quasiperiodic orbits lying ons-dimensional tori is given by [93, 315, 316]

GALIk.t/ /

8<:

constant if 2 k s1

tk�s if s < k 2N � s1

t2.k�N / if 2N � s < k 2N

: (3.54)

We note that these estimates are valid only when the corresponding conditions aresatisfied. For example, in the case of 2D maps the only possible torus is an .s D/one-dimensional invariant curve, whose tangent space is also one-dimensional. Thus, thebehavior of GALI2 (which is the only possible index in that case) is given by thethird case of (3.54), i.e. GALI2 / 1=t2, since the first two are not applicable. From(3.54) we also deduce that the behavior of GALIk for the usual case of quasiperiodicorbits lying on an N -dimensional torus is given by

GALIk.t/ /(

constant if 2 k N1

t2.k�N / if N < k 2N: (3.55)

Exercises

Exercise 3.1. (a) Use the definition of ˝ in (3.1), its properties listed in (3.2) and(3.3) to show that det ˝ D 1 and det M D ˙1. Then prove that the determinantof the 2N � 2N symplectic matrix M is exactly 1.Hint: You may find it helpful to use the theorem of polar factorization of LinearAlgebra (see [176], p. 188) to show that det M > 0.

(b) Finally, show that the eigenvalues of the matrix M are expressed as the set ofinverse pairs: �1; �2; : : : ; �N ; �N C1 D 1=�1; : : : ; �2N D 1=�N .

Exercise 3.2. Consider the n-dimensional map xkC1 D P xk; k D 0; 1; 2; : : :,where P is a constant matrix. Assume that an invertible matrix S exists such that,in the new basis xk D Syk, D D S�1PS is a diagonal matrix whose elements arethe eigenvalues of P , i.e. D D diag.�1; : : : ; �n/. Clearly, in this basis, the solution


of the problem for any initial condition y0 D S�1x0 is yk D Dky0, k D 0; 1; 2; : : : .Show that if all eigenvalues j�i j 1, i D 1; 2; : : : ; n, the map has only boundedsolutions yk , while if there is (at least) one of them satisfying j�j j > 1 the generalsolution is unbounded.

Exercise 3.3. Perform the linear transformation that uncouples (3.33) into (3.34),as described in Sect. 3.2, for the SPO1 mode under fixed boundary conditions (hint:You will have to diagonalize a tri-diagonal matrix). Next, using similar techniques,carry out the stability analysis for the other SPOs (or NNMs) of the FPU and BECHamiltonians listed at the beginning of Sect. 3.2 and derive the associated systemof uncoupled Lame equations. Are there any other such NNMs for which the N

equations of motion reduce to the solution of a single nonlinear oscillator? Are thereNNMs whose equations reduce to the solution of n D 2, 3, etc. coupled nonlinearoscillators?

Problems

Problem 3.1. Let us revisit the cubic Duffing oscillator of Exercise 2.4, in theundamped case ˛ D 0:

Pq D p; Pp D q.1 C � sin.˝t// � q3 (3.56)

(note the different signs in the second equation). Choose a small value of the drivingamplitude, e.g. � D 0:01 with ˝ D 2 and solve numerically the equations of motionto determine the periodic solutions Oqi .t/, Opi .t/, i D 1; 2 crossing the PSS (2.41) atpoints P1; P2 close to the equilibrium points .˙1; 0/ respectively of the unforced� D 0 case. For small enough � , these should be �-periodic orbits of (3.56) that arestable, in the sense that they represent elliptic points of the Poincare map (2.42).

(a) Based on your knowledge of these solutions, solve numerically thetwo-dimensional variational equations about them for initial conditions .1; 0/

and .0; 1/ to determine the 2 � 2 monodromy matrix M.�; 0/ (note that dueto the simplicity of this problem the dimensionality of the monodromy matrixcoincides with that of the Poincare map). What are the eigenvalues of M.�; 0/

and what do you conclude from them?

(b) Start increasing the value of � and repeat the previous calculation of M.�; 0/.Can you locate the first critical �c value at which the above two � periodicsolutions become unstable?

Problem 3.2. Consider the non-integrable Henon-Heiles system (2.36) withHamiltonian

H.x; y; px; py/ D 1

2.p2

x C p2y/ C 1

2.x2 C y2/ C x2y � 1

3y3 D E: (3.57)


(a) Show that this system has an exact periodic solution of the form Ox.t/ D 0,Oy.t/ D Acn.�t; �2/, with A > 0, where cn is the Jacobi elliptic cosine functionwith period T that depends on the modulus � and the constants A, �, � satisfycertain relations.

(b) Write the four-dimensional variational system of equations about this solutionand solve them numerically to determine the associated monodromy matrixM.T; 0/. Examining its eigenvalues, show that for small enough values of A

this periodic solution is linearly stable.

(c) Increasing the value of A repeat the previous calculation of M.T; 0/ and locatethe first critical value of A D A1 (and corresponding value of the energy E DE1) at which the periodic solution becomes unstable. What happens as youkeep increasing A (and the energy E)? Hint: You should find a sequence ofbifurcations at Ak > Ak�1.Ek > Ek�1/; k D 2; 3; ::: that converges to Ak !1.Ek ! 1=6/.

Problem 3.3. Continuing the work of Exercise 3.3 pursue further the stabilityanalysis of the OPM and IPM modes of the BEC Hamiltonian (3.19). Show thatthe variational equations cannot be uncoupled here as was possible for the NNMsof the FPU Hamiltonian. However, the mathematical form of the OPM and IPMsolutions and the variational equations of the BEC Hamiltonian are simple to writedown. Solve these variational equations for various choices of N D 3; 4; 5; : : :,and find the eigenvalues of the associated monodromy matrix M.T; 0/ to determinethe stability properties of the OPM and IPM solutions in terms of the values ofthe energy and norm integrals, E and F . What do you observe as the values ofthese integrals increase? Hint: Show that the IPM orbit remains stable even for largevalues of E and F , while for the OPM orbit consult Problem 3.4.

Problem 3.4. Integrate the variational equations and evaluate the monodromymatrix M.T; 0/ associated with the OPMs of the FPU and BEC Hamiltonians toshow that they become unstable, as a pair of eigenvalues of M.T; 0/ exit the unitcircle on the real axis: For FPU this happens at �1 (period-doubling bifurcation)and for BEC at C1 (pitchfork bifurcation). Fix N and examine the chaotic regionsthat arise about these orbits, as more and more pairs of eigenvalues exit the unitcircle, by computing the Lyapunov exponents in the close vicinity of these NNMs.What do you observe?

Chapter 4Normal Modes, Symmetries and Stability

Abstract The present chapter studies nonlinear normal modes (NNMs) of coupledoscillators from an altogether different perspective. Focusing entirely on periodicboundary conditions and using the Fermi Pasta Ulam ˇ (FPU�ˇ) and FPU�˛

models as examples, we demonstrate the importance of discrete symmetries inlocating and analyzing exactly a class of NNMs called one-dimensional “bushes”,depending on a single periodic function Oq.t/. Using group theoretical arguments onecan similarly identify n-dimensional bushes described by Oq1.t/; : : : ; Oqn.t/, whichrepresent quasiperiodic orbits characterized by n incommensurate frequencies.Expressing these solutions as linear combinations of single bushes, it is possibleto simplify the linearized equations about them and study their stability analytically.We emphasize that these results are not limited to monoatomic particle chains,but can apply to more complicated molecular structures in two and three spatialdimensions, of interest to solid state physics.

4.1 Normal Modes of Linear One-DimensionalHamiltonian Lattices

As the reader recalls, we began our discussion of N dof Hamiltonian systems inChap. 2 by analyzing the case of two coupled linear oscillators of equal mass m

and spring constant k described by (2.14) under fixed boundary conditions. Indeed,by a simple canonical transformation of variables (2.17), we were able to uncouplethe two equations of motion into what we called their normal mode variables andobtain the complete solution of the problem as a linear combination of normalmode oscillations with frequencies !1 D !, !2 D !

p3, ! D p

k=m. How wouldwe proceed if we were to perform the same analysis to N such coupled linearoscillators? Good question.


63

64 4 Normal Modes, Symmetries and Stability

Let us note first that the Hamiltonian function of this system

H D 1

2m

NX

j D1

p2j C k

2

NX

j D0

.qj C1 � qj /2 D E; (4.1)

leads to the equations of motion

d2qj

dt2D !2.qj �1 � 2qj C qj C1/; j D 1; 2; :::; N (4.2)

imposing again fixed boundary conditions q0.t/ � qN C1.t/ � 0, p0.t/ �pN C1.t/ � 0, with !2 D k=m. This may be called an “ordered”, or translationallyinvariant particle chain (one-dimensional lattice), since all masses and springconstants are equal. In Chap. 7 we will discuss the very important case of adisordered lattice where translational invariance is broken by a random selectionof the parameters of the system. For the time being, however, it suffices to take withno loss of generality ! D 1.

How can we uncouple equations (4.2) into a set of N single harmonic oscillators?The case N D 2 was analyzed in Chap. 2 and easily led to the transformation (2.17).Here, we are dealing with a system of N ODEs (4.2) that, when written in matrixform, involves a tridiagonal matrix S on its right hand side. Since we are lookingfor normal mode oscillations with N frequencies !q , we proceed to diagonalize thismatrix introducing new coordinates called normal mode variables Qq , Pq by thetransformation

qj Dr

2

N C 1

NX

qD1

Qq sin

�qj�

N C 1

�; pj D

r2

N C 1

NX

qD1

Pq sin

�qj�

N C 1

�;

(4.3)in terms of which (4.1) reduces to the Hamiltonian of N uncoupled harmonicoscillators

H2 D 1

2

NX

qD1

P 2q C !2

qQ2q; (4.4)

whose frequencies

!q D 2 sin

�q�

2.N C 1/

�; 1 � q � N (4.5)

are related to the eigenvalues �q of S by the simple formula �q D !2q (see (3.28)

and Exercise 4.1).Before studying this case further, let us also write down the analogous result for

the case of periodic boundary conditions qj .t/ � qj CN .t/, pj .t/ � pj CN .t/,j D 1; : : : ; N . As we ask you to demonstrate in Exercise 4.2, the reduction of(4.1) to the Hamiltonian (4.4) of N uncoupled oscillators is achieved through thetransformation

4.2 Nonlinear Normal Modes (NNMs) and the Problem of Continuation 65

Qq D 1pN

NX

j D1

�sin

2�qj

NC cos

2�qj

N

�qj ;

Pq D 1pN

NX

j D1

�sin

2�qj

NC cos

2�qj

N

�pj ;

(4.6)

and the normal mode frequencies now take the form

!q D 2 sin�q�

N

�; 1 � q � N: (4.7)

As we remark in Exercises 4.1 and 4.2, knowing the normal mode spectrain the above two cases, allows us to examine the linear (in)dependence (orcommensurability properties) of these frequencies. Remember what happened inthe case of N D 2 such oscillators in Chap. 2 under fixed boundary conditions:The two normal mode frequencies were rationally independent and the motionwas in general quasiperiodic! Does the same hold for all N ? And what about theperiodic boundary condition case, where sin.q�=N / D sin..N � q/�=N /, holdsfor q D 1; 2; : : : ; ŒN=2� (with Œ�� denoting the integer part of the argument)? Theseare important issues to which we will return later in this chapter.

4.2 Nonlinear Normal Modes (NNMs) and theProblem of Continuation

Now that we have understood the importance of normal modes in the linear case letus turn to the more difficult situation of coupled systems of nonlinear oscillators. Inparticular, we shall focus on our old friend the FPU�ˇ Hamiltonian (3.15), whichwe already met in Chap. 3 as

H D 1

2

NX

j D1

p2j C

NX

j D0

1

2.qj C1 � qj /2 C 1

4ˇ.qj C1 � qj /4 D E; (4.8)

representing an N -particle chain with quadratic and quartic nearest neighborinteractions. In Sect. 3.2, we studied some of its SPOs, under fixed and periodicboundary conditions, and found that their linear stability properties had importantconsequences for the global dynamics of the system. In particular, in the caseof fixed boundaries, we showed that the first destabilization of solutions SPO1and SPO2, corresponding to normal modes q D .N C 1/=2 and q D 2.N C 1/=3

respectively, was related to the onset of “strong” and “weak” chaos as the totalenergy E is increased.

Clearly, these SPOs are linear modes, which continue to exist as the parameter ˇ

in (4.8) is turned on and our oscillator system becomes nonlinear! Is that true? Of


course it is. Remember what we said in Sect. 3.2 about Lyapunov’s Theorem 1.3,according to which all normal modes of the linear ˇ D 0 system can be rigorouslycontinued to the nonlinear lattice, for fixed boundary conditions. This is becausethe linear frequencies (4.5) can be shown in many cases to be mutually rationallyindependent (see Exercise 4.1). Unfortunately, Lyapunov’s theorem does not applyto the FPU problem under periodic boundary conditions.

So what do we do in that case? Don’t worry. We will construct its NNMs later inthis chapter using discrete symmetries of the equations of motion and discuss theirstability properties in detail. For the time being let us complete our discussion ofthe NNMs of the FPU�ˇ system under fixed boundaries. This is indeed a veryimportant problem as these NNMs are intimately connected with the historicalparadox of the FPU recurrences, which we shall discuss in detail in Chap. 6.

Note that substituting (4.3) into (4.8) allows us to write the FPU�ˇ Hamiltonianin the form H D H2 C H4 in which the quadratic part corresponds to N uncoupledharmonic oscillators, as in (4.4) above. On the other hand, the quartic part of theHamiltonian becomes

H4 D ˇ

8.N C 1/

NX

q;l;m;nD1

Cq;l;m;n!q!l !m!nQqQlQmQn; (4.9)

where the coefficients Cq;l;m;n take non-zero values only for particular combinationsof the indices q; l; m; n, namely

Cq;l;m;n DX

˙

�ıq˙l˙m˙n;0 � ıq˙l˙m˙n;˙2.N C1/

�(4.10)

in which all possible combinations of the ˙ signs arise, with ıij D 1 for i D j andıij D 0 for i ¤ j . Thus, in the new canonical variables, the equations of motion areexpressed as

RQq C !2qQq D � ˇ

2.N C 1/

NX

l;m;nD1

Cq;l;m;n!q!l!m!nQlQmQn: (4.11)

If ˇ D 0, the individual harmonic energies Eq D .P 2q C !2

qQ2q/=2 are preserved

since they constitute a set of N integrals in involution. When ˇ ¤ 0, however, theharmonic energies become functions of time and only the total energy E , (4.8), isconserved. One may thus define a specific energy of the system as " D E=N , whilethe average harmonic energy of each mode over a time interval 0 � t � T is givenby the integral NEq.T / D 1

T

R T

0 Eq.t/dt.As we discuss in Chap. 6, in classical FPU experiments, one starts with the total

energy distributed among a small subset of linear modes. Then, as every physicsstudent knows, equilibrium statistical mechanics predicts that, due to nonlinearinteractions, after a short time interval the energy of all non-excited modes will

4.3 Periodic Boundary Conditions and Discrete Symmetries 67

grow and a kind of equipartition will occur with the energy being shared equally byall modes, i.e.

limT !1

NEq.T / D " ; q D 1; : : : ; N: (4.12)

But this does not happen in the FPU system! At least for a range of low E values,one finds that the total energy returns periodically to the originally excited modes,yielding the famous FPU recurrences which persist even when T becomes verylarge. The paradox lies in the fact that such deviations from equipartition were notexpected to occur in a nonlinear and non-integrable Hamiltonian system as the FPU.Is it still a paradox to date? Not any longer, we claim. As will be explained inChap. 6, the dynamics of FPU recurrences can be understood in terms of exponentiallocalization of periodic and quasiperiodic solutions in q-modal space.

4.3 Periodic Boundary Conditions and Discrete Symmetries

Let us recall from the above discussion that, when periodic boundary conditions areimposed, i.e. qj D qj CN , pj D pN Cj , j D 1; 2; : : : ; N , the linear mode spectrumof the FPU-ˇ particle chain becomes degenerate for all N and Lyapunov’s theoremcannot be invoked. How do we proceed to study existence and stability of NNMsin that case? As we explain below, this is a situation where the identification ofthe system’s discrete symmetries turns out to be very helpful in the analysis of thedynamics.

In the present chapter we shall demonstrate how one may use the powerfultechniques of group theory to establish the existence of such NNMs for a variety ofmechanical systems, including particle chains in one dimension, as well as certaintwo-dimensional and three-dimensional structures of interest to solid state physics.These NNMs are simple periodic orbits, which we shall call one-dimensionalbushes. We will then show how one can combine such periodic solutions toform multi-dimensional bushes of NNMs and exploit symmetries to simplify thevariational equations about them and study their (linear) stability. Ultimately, ofcourse, one would like to be able to obtain invariant manifolds on which thedynamics of the system is as simple as possible. This is not easy to do in general,but if the equations of motion possess discrete symmetry groups, one can single outsome of these manifolds using group theoretical methods developed in [81,82,295](see also [64] for a recent review).

4.3.1 NNMs as One-Dimensional Bushes

Let us illustrate the main steps of the bush theory on the FPU-ˇ Hamiltonian system(4.8). The ODEs describing the longitudinal vibrations of the FPU-ˇ chain can bewritten in the form


Rqi D f .qiC1 � qi / � f .qi � qi�1/; i D 1; : : : ; N; (4.13)

where qi .t/ is the displacement of the i th particle from its equilibrium state at time t ,while the force f .�q/ depends on the spring deformation �q as follows:

f .�q/ D �q C ˇ.�q/3: (4.14)

We also assume that ˇ > 0 and impose periodic boundary conditions. Thus,we may study the dynamics of this chain by attempting to solve (4.13) for the“configuration” vector

X.t/ D fq1.t/; q2.t/; :::; qN .t/g; (4.15)

whose components are the individual particle displacements.As is well-known, it is pointless to try to obtain this vector as a general solution

of (4.13). We, therefore, concentrate on studying special solutions represented bythe NNMs described in the previous section. In particular, let us begin with thefollowing simple periodic solution

X.t/ D f Oq.t/; � Oq.t/; Oq.t/; � Oq.t/; : : : ; Oq.t/; � Oq.t/g; (4.16)

which is easily seen to exist in the FPU�ˇ chain with an even number ofparticles (N mod 2 D 0). This solution is called the OPM or “�-mode” and is fullydetermined by only one arbitrary function Oq.t/ as we already discovered in Chap. 3.It represents an example of the type of NNMs introduced in [286] and expresses anexact dynamical state which can be written in the form

X.t/ D Oq.t/f1; �1; j1; �1; j : : : ; j1; �1g: (4.17)

Hence, the following question naturally arises: Are there any other such exactNNMs in the FPU-ˇ chain? Some examples of such modes are already known in theliterature [13,203,218,272,283,303,304,351], under various terminologies. Belowwe list these exact states in detail

X.t/ D Oq.t/f1; 0; �1; j1; 0; �1; j; : : :g; !2 D 3; .N mod 3 D 0/; (4.18)

X.t/ D Oq.t/f1; �2; 1; j1; �2; 1; j; : : :g; !2 D 3; .N mod 3 D 0/; (4.19)

X.t/ D Oq.t/f0; 1; 0; �1; j0; 1; 0; �1; j; : : : :g; !2 D 2; .N mod 4 D 0/;

(4.20)X.t/ D Oq.t/f1; 1; �1; �1; j1; 1; �1; �1j; : : :g; !2 D 2; .N mod 4 D 0/;

(4.21)

X.t/ D Oq.t/f0; 1; 1; 0; �1; �1; j0; 1; 1; 0; �1; �1j; : : :g; !2 D 1;

.N mod 6 D 0/: (4.22)

4.3 Periodic Boundary Conditions and Discrete Symmetries 69

Let us comment on certain properties of the above NNMs: Note that in (4.17)–(4.22) we have identified for each of these NNMs a primitive cell (divided byvertical lines), whose number of elements m is called the multiplicity number.These numbers have the values: m D 2 for (4.17), m D 3 for (4.18) and (4.19),m D 4 for (4.20) and (4.21) and m D 6, for (4.22). Observe also that, due to theperiodicity of the boundary conditions, the FPU-ˇ chain in its equilibrium state isinvariant under translation by the inter-particle distance a, while in its vibrationalstates it is invariant under a translation by ma. Moreover, NNMs (4.17)–(4.22) differnot only by translational symmetry (ma), but also by some additional symmetrytransformations which are discussed below.

Each of the NNMs (4.17)–(4.22) depends on only one function Oq.t/ and is,therefore, said to describe a one-dimensional “dynamical domain”, borrowing theterm from the theory of phase transitions in crystals. For example, consider the mode(4.18): It is easy to check that cyclic permutations of each primitive cell producesother modes, which differ only by the position of their stationary particles. As aconsequence, these NNMs possess equivalent dynamics and, in particular, turn outto share the same stability properties. Thus, we only need to study one representativemember of each set.

Many aspects of existence and stability of NNMs have been discussed in theliterature [13, 66, 85, 87, 203, 218, 272, 283, 298, 303, 304, 351]. What is importantfrom our point of view is to pose certain fundamental questions concerning theseNNMs, which we shall proceed to answer in the sections that follow:

(Q1) Is the list (4.17)–(4.22) of NNMs for the FPU-ˇ chain complete? Indeed, atfirst sight, it seems that many other NNMs exist, for example, modes whosemultiplicity number m is different from those listed above.

(Q2) What kind of NNMs arise in nonlinear chains with different interactions thanthose of the FPU-ˇ chain? In most papers (see [203, 272, 303, 304, 351])the NNMs listed above are discussed by analyzing dynamical equationsconnected only with FPU-ˇ inter-particle interactions.

(Q3) Do there exist NNMs for Hamiltonian systems that are more complicated thanmonoatomic chains? For example, can one pose this question for diatomicnonlinear chains (with particles having alternating masses), two-dimensional(2D) lattices, or 3D crystal structures?

(Q4) Can one construct exact multi-dimensional bushes in nonlinear N -particleHamiltonian systems and study their stability by locating the unstable man-ifolds of orbits characterized by a number of different frequencies? Aswe shall discover in subsequent sections, there exist interesting domainsof regular motion depending on more than one frequency and character-ized by families of quasiperiodic functions, which form s-dimensional tori,with s � 2.


4.3.2 Higher-Dimensional Bushes and Quasiperiodic Orbits

Let us consider the following exact solution of the FPU-ˇ chain with N mod 6 D 0:

X.t/ D f0; Oq1.t/; Oq2.t/; 0; � Oq2.t/; � Oq1.t/j : : : j0; Oq1.t/; Oq2.t/; 0; � Oq2.t/; � Oq1.t/jg:(4.23)

This bush has a multiplicity number m D 6 and is defined by two functions, Oq1.t/

and Oq2.t/, satisfying a system of two nonlinear autonomous ODEs. Thus, it isan example of what we call a two-dimensional bush that represents an exactquasiperiodic motion involving two frequencies.

We remark at this point that, in the framework of bush theory, the problem offinding invariant manifolds of exact solutions in a physical system with discretesymmetry can be solved without any information about the inter-particle forces.However, the explicit form of the bush dynamical equations does depend onthe inter-particle interactions of the system under consideration. In the sectionsthat follow, we will show that the theory of bushes gives definite and quitegeneral answers to the above questions Q1–Q3 in nonlinear systems with discretesymmetries. We shall then proceed to answer question Q4, using the analytical andnumerical results described in Chap. 6.

More specifically, we shall demonstrate that it is possible to use Poincare-Linstedt series expansions to construct exact quasiperiodic solutions with s � 2

frequencies, which belong to the lower part of the normal mode spectrum, withq D 1; 2; 3; : : : . Thus, we will be able to establish the important property that theenergies Eq excited by these low-dimensional, so-called q-tori, are exponentiallylocalized in q-space [94]. The relevance of this fact becomes apparent when westudy the well-known phenomenon of FPU recurrences, which are remarkablypersistent in FPU chains (with fixed or periodic boundary conditions), at low valuesof the total energy E .

We will also study in Chap. 6 the stability of these q-tori as accurately as possible,using methods that do not rely on Floquet theory. In particular, we shall exploit theapproach of the GALIk indicators k D 2; : : : ; 2N , described in Chaps. 3 and 5,to estimate the destabilization threshold at which q-tori are destroyed, leading tothe breakdown of FPU recurrences and the eventual onset of energy equipartitionamong all modes.

4.4 A Group Theoretical Study of Bushes

Let us begin with the case of NNMs, which represent one-dimensional bushesof orbits that help to determine all symmetry groups of the equations describingthe vibrations of a given mechanical system. The set of these groups constitutesthe parent symmetry group of all transformations that leave the given system ofequations invariant. Let us consider, for example, the dynamical equations (4.13)

4.4 A Group Theoretical Study of Bushes 71

with (4.14) for the FPU-ˇ chain with periodic boundary conditions and an evennumber of particles. Clearly, these remain invariant under the action of an operator Oathat shifts it by the lattice spacing a. This operator generates the translational group

T D fOe; Oa; Oa2; : : : ; OaN �1g; OaN D Oe; (4.24)

where Oe is the identity element and N is the order of the cyclic group T . Theoperator Oa induces a cyclic permutation of all particles of the chain and, therefore,acts on the configuration vector X.t/ of (4.15) as follows

OaX.t/ � Oafq1.t/; q2.t/; : : : ; qN �1.t/; qN .t/g DfqN .t/; q1.t/; q2.t/; : : : ; qN �1.t/g: (4.25)

Another important element of the symmetry group of transformations of themonoatomic chain is the inversion Oi with respect to the center of the chain, whichacts on the vector X.t/ in the following way

OiX.t/ � Oifq1.t/; q2.t/; : : : ; qN �1.t/; qN .t/g Df�qN .t/; �qN �1.t/; : : : ; �q2.t/; �q1.t/g: (4.26)

The complete set of symmetry transformations, of course, includes also all productsOak Oi of the pure translations Oak (k D 1; 2; : : : ; N � 1) with the inversion Oi and formsthe so-called dihedral group D which can be written as a direct sum of the twocosets T and T � Oi

D D T ˚ T � Oi : (4.27)

This is a non-Abelian group induced by two generators ( Oa and Oi ) through thefollowing generating relations

OaN D Oe; Oi 2 D Oe; Oi Oa D Oa�1Oi : (4.28)

Clearly, operators Oa and Oi induce the following changes of variables

Oa W q1.t/ ! qN .t/; q2.t/ ! q1.t/; : : : ; qN .t/ ! qN �1.t/IOi W q1.t/ $ �qN .t/; q2.t/ $ �qN �1.t/; q3.t/ $ �qN �2.t/; : : : :

(4.29)

It is now straightforward to check that upon acting on (4.13) with transformation(4.29) the system is transformed to an equivalent form. Moreover, since (4.13) areinvariant under the actions of Oa and Oi , they are also invariant with respect to allproducts of these two operators and, therefore, the dihedral group D is indeeda symmetry group of (4.13) for a monoatomic chain with arbitrary inter-particleinteractions. As a consequence, the dihedral group D can be considered as a parentsymmetry group for all monoatomic nonlinear chains as, for example, the FPU-˛chain, whose inter-particle interactions are characterized by the force

f .�q/ D �q C ˛.�q/2: (4.30)


Of course, since in the case of the FPU-ˇ chain the force is an odd function ofits argument, the FPU-ˇ chain possesses a higher symmetry group than the FPU-˛model. Indeed, let us introduce the operator Ou which changes the signs of all atomicdisplacements without their transposition

OuX � Oufq1.t/; q2.t/; : : : ; qN �1.t/; qN .t/g D f�q1.t/; �q2.t/; : : : ;

� qN �1.t/; �qN .t/g: (4.31)

It can be easily checked that the operator Ou generates a transformation of all thevariables qi .t/, i D 1; : : : ; N in (4.13), (4.14), which leads to an equivalent form ofthese equations. Therefore, the operator Ou and all its products with elements of thedihedral group D belong to the full symmetry group of the FPU-ˇ chain. Clearly,the operator Ou commutes with all the elements of the dihedral group D and thus wecan consider the group

G D D ˚ D � Ou (4.32)

as the parent symmetry group of the FPU-ˇ chain. The group G contains twice asmany elements as the dihedral group D and, therefore, possesses a greater numberof subgroups.

Finally, let us note that it is sufficient for our purposes to define any symmetrygroup by listing only its generators which we denote by square brackets, forexample, we write T Œ Oa�, DŒ Oa; Oi �, GŒ Oa; Oi ; Ou�. The complete set of group elementsis written in curly brackets, as in (4.24).

4.4.1 Subgroups of the Parent Group and Bushes of NNMs

Let us consider now a specific configuration vector X.j /.t/, see (4.15), whichdetermines a displacement pattern at time t , and let us act on it successively bythe operators Og that correspond to all the elements of a parent group G. The fullset Gj of elements of the group G under which X.j /.t/ turns out to be invariantgenerates a certain subgroup of G (Gj � G). We then call X.j /.t/ invariant underthe action of the subgroup Gj of the parent group G and use it to determine the bushof NNMs corresponding to this subgroup.

Thus, in the framework of this approach, one must list all the subgroups of theparent group G to obtain all the bushes of NNMs of different types. This can be doneby standard group theoretical methods. In [87] a simple crystallographic techniquewas developed for singling out all the subgroups of the parent group of anymonoatomic chain, following the approach of a more general method [80, 83, 84].Here, we demonstrate how one can obtain bushes of NNMs if the subgroups arealready known.

Let us consider the subgroups Gj of the dihedral group D. Each Gj containsits own translational subgroup Tj � T , where T is the full translational group(4.24). If N is divisible by 4, say, there exists a subgroup T4 D Œ Oa4� of the group


T D Œ Oa�. If a vibrational state of the chain possesses the symmetry group T4 DŒ Oa4� � fOe; Oa4; Oa8; : : : ; OaN �4g, the displacements of atoms that lie at a distance 4a

from each other in the equilibrium state turn out to be equal, since the operator Oa4

leaves the vector X.t/ invariant.For example, for the case N D 12, the operator Oa4 permutes the coordinates

of X D fq1; q2; : : : ; q12g taken in quadruplets .qi ; qiC1; qiC2; qiC3/, i D 1; 5; 9,while from equation Oa4X.t/ D X.t/ one deduces qi D qiC4, i D 1; 2; 3; 4. Thus,the vector X.t/ contains three times the quadruplets q1; q2; q3; q4, where qi .t/ .i D1; 2; 3; 4/ are arbitrary functions of time and can be written as follows

X.t/ D f q1.t/; q2.t/; q3.t/; q4.t/ j q1.t/; q2.t/; q3.t/; q4.t/ j q1.t/; q2.t/; q3.t/; q4.t/ g:(4.33)

In other words, the complete set of atomic displacements can be divided into N=4

(in our case, N=4 D 3) identical subsets, which are called extended primitivecells. In the bush (4.33), the extended primitive cell contains four atoms and thevibrational state of the whole chain is described by three such cells. Thus, theextended primitive cell for the vibrational state with the symmetry group T4 D Œ Oa4�

has size equal to 4a, which is four times larger than the primitive cell of the chainat the equilibrium state.

It is essential that some symmetry elements of the dihedral group D disappearas a result of the symmetry reduction D D Œ Oa; Oi � ! T4 D Œ Oa4�. There are fourother subgroups of the dihedral group D, corresponding to the same translationalsubgroup T4 D Œ Oa4�, namely

Œ Oa4; Oi �; Œ Oa4; OaOi �; Œ Oa4; Oa2Oi �; Œ Oa4; Oa3Oi �: (4.34)

Each of these subgroups possesses two generators, namely Oa4 and an inversionelement Oak Oi .k D 0; 1; 2; 3/. Note that the Oak Oi differ from each other by the positionof the center of inversion. Subgroups Œ Oa4; Oak Oi � with k > 3 are equivalent to thoselisted in (4.34), since the second generator Oak Oi can be multiplied from the left byOa�4, representing the inverse element with respect to the generator Oa4. Thus, thereexist only five subgroups of the dihedral group (with N mod 4 D 0) constructed onthe basis of the translational group T4 D Œ Oa4�: This one and the four listed in (4.34).

Now, let us examine the bushes corresponding to the subgroups (4.34). Thesubgroup Œ Oa4; Oi � consists of the following six elements

Oe; Oa4; Oa8; Oi ; Oa4Oi ; Oa8Oi � Oi Oa4: (4.35)

The invariance of X.t/ with respect to this group can be written as follows:

Oa4X.t/ D X.t/; OiX.t/ D X.t/; (4.36)

while the invariance of the vector X.t/ under the action of the group generators Œ Oa4�

and ŒOi � guarantees its invariance under all elements of this group.


As explained above, the equation Oa4X.t/ D X.t/ is satisfied by the vector X.t/

(see (4.33)), while OiX.t/ D X.t/ also holds, from which we obtain the followingrelations: q1.t/ D �q4.t/, q2.t/ D �q3.t/. Therefore, for N D 12, the invariantvector X.t/ of the group [ Oa4; Oi ] can be written in the form

X.t/ D fq1.t/; q2.t/; �q2.t/; �q1.t/ˇˇq1.t/; q2.t/; �q2.t/; �q1.t/

ˇˇq1.t/; q2.t/; �q2.t/; �q1.t/g; (4.37)

where q1.t/ and q2.t/ are arbitrary functions of time.Thus, the subgroup [ Oa4; Oi ] of the dihedral group D generates a two-dimensional

bush of NNMs. The explicit form of the differential equations governing the twovariables q1.t/ and q2.t/ can now be obtained by substitution of the ansatz (4.37)into the FPU-ˇ equations (4.13), (4.14). We shall, hereafter, denote the bush (4.37)in the form

BŒ Oa4; Oi � D ˇq1; q2; �q2; �q1

ˇ; (4.38)

showing the atomic displacements in only one extended primitive cell and omittingthe argument t in the variables q1.t/, q2.t/.

Proceeding in a similar manner, we obtain bushes of NNMs for the other threegroups listed in (4.34)

BŒ Oa4; OaOi � D ˇˇ0; q; 0; �q

ˇˇ; (4.39)

BŒ Oa4; Oa2Oi � D ˇˇq1; �q1; q2; �q2

ˇˇ; (4.40)

BŒ Oa4; Oa3Oi � D ˇˇq; 0; �q; 0

ˇˇ: (4.41)

More generally, we conclude that for sufficiently large extended primitive cellsit will not be possible to find enough symmetry elements to give rise to NNMs,since the bushes of the corresponding displacement patterns are multi-dimensional.For this reason, there exists only a very specific number of bushes for any fixeddimension beyond the (one-dimensional) NNMs!

4.4.2 Bushes in Modal Space and Stability Analysis

Recall now that our vectors X.t/, giving rise to bushes of NNMs, are defined inthe configuration space R

N . If we now introduce in this space a basis set of vectorsf'1; '2; : : : ; 'N g, we can represent the dynamical regime of our mechanical systemby a linear combination of the vectors 'j with time dependent coefficients �j .t/ asfollows

X.t/ DNX

j D1

�j .t/'j ; (4.42)

where the functions �j .t/ entering this decomposition may be thought of as newdynamical variables.


Let us observe that every term in the sum (4.42) has the form of a NNM,whose basis vector 'j determines a displacement pattern, while the functions�j .t/ determine the time evolution of the atomic displacements. Because of thisinterpretation, we can consider a given dynamical regime X.t/ as a bush of NNMs.In fact, we may also speak of root modes and secondary modes of a given bush (seebelow).

Note that each term �j .t/'j in the sum (4.42) is not, in general, a solution of thedynamical equations of the considered mechanical system, while a specific linearcombination of a number of these modes can represent such a solution and thusdescribe an exact dynamical regime. Sometimes, for brevity, we will use the word“mode” not only for the term �j .t/'j , but also for �j .t/ itself.

We now write our Hamiltonian as the sum of a kinetic energy and a potentialenergy V.X/ part, and assume that V.X/ can be decomposed into a Taylor serieswith respect to the atomic displacements qi .t/ from their equilibrium positionsand all terms whose orders are higher than 2 are neglected. As a result, Newton’sequations

mi Rqi D � @V

@qi

; .i D 1; : : : ; N / (4.43)

are linear differential equations, with constant coefficients. As we discussed inChap. 2, each normal mode is a particular solution to (4.43) of the form

X.t/ D c cos.!t C '0/; (4.44)

where the N -dimensional constant vector c D fc1; c2; : : : ; cN g and the constantphase �0 are determined by the initial displacements of all particles from theirequilibrium state and ! is the normal mode frequency.

Since V.X/ has a quadratic form, substituting (4.44) into (4.43) and eliminatingcos.!t C'0/ from the resulting equations reduces the problem of finding the normalmodes to the task of evaluating the eigenvalues and eigenvectors of the matrix K

with coefficients

kij D @2V

@qi @qj

ˇˇˇˇXD0

; i; j D 1; : : : ; N: (4.45)

Since the the N � N matrix K is real and symmetric, it possesses N eigenvectorscj (j D 1; : : : ; N ) and N eigenvalues !2

j . The complete collection of these vectors,i.e. the N normal coordinates, can be used as the basis of our configuration space,hence we may write

X.t/ DNX

iD1

�j .t/cj ; (4.46)

where X.t/ D fq1.t/; q2.t/; : : : ; qN .t/g, while �j .t/ are the new dynamical vari-ables that replace the old variables qi .t/ .i D 1; : : : ; N /.


If the transformation to normal coordinates is used in the absence of degenera-cies, the corresponding system of linear ODEs leads to a set of uncoupled harmonicoscillators

R�j .t/ C !2j �j .t/ D 0; j D 1; : : : ; N;

with the well-known solution

�j .t/ D aj cos�!j t C '0j

�; (4.47)

where aj and '0j are arbitrary constants.Note the distinction we make between a normal coordinate, represented by the

eigenvector cj , and a normal mode, referring to the product of the vector cj and thetime-periodic function �j .t/ D cos

�!j t C '0j

�.

Let us begin by considering individual NNMs, representing one-dimensionalbushes of the FPU-ˇ Hamiltonian. To study bushes of NNMs in monoatomic chains,we shall choose the complete set of normal coordinates 'k as the basis of ourconfiguration space. Here, we use the normal coordinates in the form presentedin [272]:

'k D�

1pN

sin

�2�k

Nn

�C cos

�2�k

Nn

�ˇˇˇ n D 1; : : : ; N

�; k D 0; : : : ; N �1;

(4.48)where the subscript k refers to the mode and the subscript n refers to the specificparticle. The vectors 'k , k D 0; 1; 2; : : : ; N � 1, form an orthonormal basis, inwhich we can expand the set of atomic displacements X.t/ corresponding to a givenbush as follows

X.t/ DN �1X

kD0

�k.t/'k (4.49)

(see (4.42)). For example, one obtains in this way the following expressions for thebushes BŒ Oa4; Oi � and BŒ Oa4; Oa2Oi � (see (4.38) and (4.40))

BŒ Oa4; Oi � W X.t/ D f q1.t/; q2.t/; �q1.t/; �q2.t/ j q1.t/; q2.t/;

� q1.t/; �q2.t/ j : : : gD �.t/'N=2 C �.t/'3N=4 ; (4.50)

BŒ Oa4; Oa2Oi � W X.t/ D e�.t/'N=2 Ce�.t/’N=4: (4.51)

From the complete basis (4.48) only the vectors

'N=2 D 1pN

.�1; 1; �1; 1; �1; 1; �1; 1; �1; 1; �1; 1; : : :/; (4.52)

'N=4 D 1pN

.1; �1; �1; 1; 1; �1; �1; 1; 1; �1; �1; 1; : : :/; (4.53)


'3N=4 D 1pN

.�1; �1; 1; 1; �1; �1; 1; 1; �1; �1; 1; 1; : : :/; (4.54)

contribute to the two-dimensional bushes (4.50), (4.51), which are equivalent toeach other and constitute examples of dynamical domains. For the bush BŒ Oa4; Oi �, wefind the following relations between the old variables q1.t/, q2.t/ (corresponding tothe configuration space) and the new ones �.t/, �.t/ (corresponding to the modalspace)

�.t/ D �p

N2

Œq1.t/ � q2.t/�;

�.t/ D �p

N2

Œq1.t/ C q2.t/�:(4.55)

Thus, each of the above bushes consists of two modes. One of these modesis the root mode ('3N=4 for the bush BŒ Oa4; Oi � and 'N=4 for the bush BŒ Oa4; Oa2Oi �),while the other mode 'N=2 is the secondary mode. Indeed, according to [81,82,294]the symmetry of the secondary modes must be higher or equal to the symmetry ofthe root mode. In our case, as we deduce from (4.53), the translational symmetryof the mode 'N=4 is Oa4 (acting by this element on (4.53) produces the samedisplacement pattern), while the translational symmetry of the mode 'N=2 is Oa2,which has twice as many symmetry elements as that of 'N=4 (see (4.52) and (4.53)).

Note that the full symmetry of the modes 'N=4 and 'N=2 is represented by Œ Oa4; Oa2Oi �and Œ Oa2; Oi �, respectively.

Let us now return to the Hamiltonian of the FPU system written as follows

H D T C V D 1

2

NX

nD1

Pq2n C 1

2

NX

nD1

.qnC1 � qn/2 C

p

NX

nD1

.qnC1 � qn/p: (4.56)

Here p D 3, D ˛ for the FPU-˛ chain, and p D 4, D ˇ for the FPU-ˇ chain,while T and V are the kinetic and potential energies, respectively. We assume againperiodic boundary conditions.

Let us consider the set of atomic displacements corresponding to the two-dimensional bush BŒ Oa4; Oi �

X.t/ D f x; y; �y; �x j x; y; �y; �x j x; y; �y; �x j : : : g; (4.57)

where we rename q1.t/ and q2.t/ from (4.50) as x.t/ and y.t/, respectively.Substituting the particle displacements from (4.57) into the Hamiltonian (4.56),choosing in this case the FPU-˛ chain, we obtain the kinetic and potential energies

T D N

4. Px2 C Py2/; (4.58)

V D N

4.3x2 � 2xy C 3y2/ C N˛

2.x3 C x2y � xy2 � y3/: (4.59)


These expressions are valid for an arbitrary FPU-˛ chain with N mod 4 D 0. Thesize of the extended primitive cell for the vibrational state (4.57) is equal to 4a

and, therefore, when calculating the energies T and V , we may restrict ourselves tosumming over only one such cell. In the present case, Newton’s equations become

� Rx C .3x � y/ C ˛.3x2 C 2xy � y2/ D 0;

Ry C .3y � x/ C ˛.x2 � 2xy � 3y2/ D 0:(4.60)

Let us emphasize that these equations do not depend on the number N of theparticles in the chain, only N mod 4 D 0 must hold.

Equations 4.60 are written in terms of the particle displacements x.t/ and y.t/.From them, it is easy to obtain the dynamical equations for the bush in terms of thenormal modes �.t/ and �.t/. Using the relations (4.55) between the old and newvariables, we find the following equations for the bush BŒ Oa4; Oi � in the modal space

R� C 4� � 4˛pN

�2 D 0; (4.61)

R� C 2� � 8˛pN

�� D 0: (4.62)

Thus, the Hamiltonian for the bush BŒ Oa4; Oi �, considered as a two-dimensionaldynamical system, can be written in the modal space as follows

HŒ Oa4; Oi � D 1

2. P�2 C P�2/ C .2�2 C �2/ � 4˛p

N��2: (4.63)

The stability of bushes of modes was discussed, in general, in [81,82,294], whilein the case of the FPU chains it was considered in [85, 87]. Following these papers,we also discuss here the stability of a given bush of normal modes with respect toits interactions. Note that there is an essential difference between the interactions ofthe modes that belong and those that do not belong to a given bush: In the formercase, we speak of a “force interaction” and in the latter of a “parametric interaction”(see [81, 82]). Let us illustrate this with an example.

As was shown above, the two-dimensional bush BŒ Oa4; Oi � for the FPU-˛ chain isdescribed by (4.61) and (4.62). These equations admit a special solution of the form

�.t/ ¤ 0; �.t/ � 0: (4.64)

which can be excited by imposing the initial conditions: �.t0/ D �0 ¤ 0, P�.t0/ D 0,�.t0/ D 0, P�.t0/ D 0. Substitution of (4.64) into (4.61) produces the dynamicalequation of the one-dimensional bush BŒ Oa2; Oi � (see [87]) consisting of only onemode �.t/

R� C 4� D 0; (4.65)

with the simple solution


�.t/ D �0 cos.2t/; (4.66)

taking (with no loss of generality) the initial phase to be equal to zero. Substituting(4.66) into (4.62), we obtain

R� C2 � 8˛�0p

Ncos.2t/

� D 0 ; (4.67)

which is easily transformed into the standard form of the Mathieu equation [6]

R� C Œa � 2q cos.2t/�� D 0: (4.68)

As is well-known, there exist domains of stable and unstable motion of theMathieu equation (4.68) in the .a; q/ plane of its parameters [6]. The one-dimensional bush BŒ Oa2; Oi � is stable for sufficiently small amplitudes �0 of the mode�.t/, but becomes unstable when �0 > 0 is increased . This phenomenon, similarto the well-known parametric resonance, takes place at �0 values which lie withinthe domains of unstable motion of Mathieu’s equation (4.67).

The loss of stability of the dynamical regime (4.64) (representing the bushBŒ Oa2; Oi �), manifests itself in the exponential growth of the mode �.t/, which wasidentically zero for the vibrational state (4.64) and oscillatory for the stable regimeof (4.68). As a result of �.t/ ¤ 0, the dimension of the original one-dimensionalbush BŒ Oa2; Oi � increases and the bush is transformed into the two-dimensional bushBŒ Oa4; Oi �. This is accompanied by a breaking of the symmetry of the vibrational state(the symmetry of the bush BŒ Oa2; Oi � is twice as high as that of the bush BŒ Oa4; Oi �).

In general, we may view a given bush as a stable dynamical object if the completecollection of its modes (and hence also its dimension) do not change in time.All other modes of the system possess zero amplitudes and are therefore called“sleeping” modes. If we increase the intensity of bush vibrations, some sleepingmodes (because of parametric interactions with the active modes [81, 82]) canlose their stability and become excited. In this situation, we speak of the loss ofstability of the original bush, since the dimension of the vibrational state (the numberof active modes) becomes larger, while its symmetry becomes lower. Thus, as aconsequence of stability loss, the original bush transforms into another bush ofhigher dimension.

Note, however, that the above destabilization of the (one-dimensional) bushBŒ Oa2; Oi � with respect to its transformation into the (two-dimensional) bush BŒ Oa4; Oi �,falls within the framework of Floquet theory and the analysis of a simple Mathieu’sequation. In the case of a more general perturbation of the N particle chain, one mustexamine the stability of the bush BŒ Oa4; Oi � with respect to all other modes, as well.This cannot be done using Floquet theory, since the variations that need to be studiedare not about a periodic but a quasiperiodic orbit with two rationally independentfrequencies. In Chap. 6, the stability analysis of a multi–dimensional bush is


described theoretically and is performed efficiently and accurately employing theGALI method described in Chap. 5.

Indeed, as we shall show in Chap. 6, it is possible that tori continue to existat energies higher than the thresholds (obtained by Floquet theory) where their“parent” modes become linearly unstable! This ensures stability over a widerdomain than just an infinitesimal neighborhood of the simple periodic orbitsrepresenting the NNM’s of the FPU chain.

4.5 Applications to Solid State Physics

As is well-known, the dynamics of Hamiltonian systems is a broad field with a widerange of applications. Of course, in this book the emphasis is mainly placed oncoupled oscillators of N dof that one encounters primarily in classical mechanics.However, as we have already seen in Chap. 3, with regard to N –particle one-dimensional lattices, when N becomes arbitrarily large, it is possible to examinecertain very important issues of interest to statistical mechanics. Still, there aresome basic questions concerning these Hamiltonian systems that may be posed moregenerally. One such question addresses the existence and stability of NNMs, whichare the most natural states to consider in certain problems of atomic or molecularvibrations.

In the previous sections of this chapter, we described a convenient way to studyNNMs, as bushes of orbits in systems which possess discrete symmetries, usingsome basic concepts of group theory. From this viewpoint, one realizes that thisapproach might perhaps be useful for the study of more general mechanical systemsoccurring in solid state physics. This expectation indeed turns out to be fulfilled,as we will discuss in this section, in the examples of a square molecule in twodimensions and an octahedral molecule in three dimensions.

4.5.1 Bushes of NNMs for a Square Molecule

Let us consider, therefore, a square molecule represented by a mechanical systemwhose equilibrium state is shown in Fig. 4.1. The four atoms of this molecule areshown as filled circles at the vertices of the square, while the number of every atomand its .x; y/ coordinates are also given in the figure.

If we suppose that the four atoms can oscillate about their equilibrium positionsonly in the .x; y/ plane, we immediately realize that this model can be describedby a Hamiltonian system possessing eight dof. Furthermore, we will not assumeat this stage any specific type of inter-particle interactions, so that we may treatbushes of NNMs of this system as purely geometrical objects. The equilibriumconfigurations of our molecule possess a symmetry group denoted by C4v below,

4.5 Applications to Solid State Physics 81

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12

3 4

x

y

.�1; 1/ .1; 1/

.�1; �1/ .1; �1/

Fig. 4.1 The model of a square molecule with one atom at each of its four vertices (after [64])

according to which, at equilibrium, the system is invariant under the action of thefollowing transformations

1. Rotations by the angles 0ı, 90ı, 180ı, 270ı about a z axis orthogonal to the planeof Fig. 4.1 passing through the center of the square. We denote these rotations byg1, g2, g3, g4, respectively.

2. Reflections across four mirror planes orthogonal to the plane of the moleculecontaining the z axis and shown by bold lines in Fig. 4.1. Two of them are“coordinate” planes (g5, g7) and the other two are “diagonal” planes (g6, g8).

The above symmetry elements can be explicitly defined as follows

g1.x; y/ D .x; y/; g2.x; y/ D .�y; x/; g3.x; y/ D .�x; �y/; g4.x; y/ D .y; �x/

g5.x; y/ D .�x; y/; g6.x; y/ D .�y; �x/; g7.x; y/ D .x; �y/; g8.x; y/ D .y; x/:(4.69)

Thus, the symmetry group of the square molecule, C4v, contains eight elementsg1; : : : ; g8, determined by (4.69). This group is non-Abelian since, e.g., g2 �g8 D g5,while g8 � g2 D g7. According to Lagrange’s theorem, the order of any subgroup isa divisor of the order of the full group. Therefore, for the case of the group G D C4v

with the order m D 8, there exist only four subgroups Gj with order equal tom D 1; 2; 4; 8 as follows


m D 1 W G1 D fg1g D C1

m D 2 W G2 D fg1; g3g D C2

G3 D fg1; g5g and G03 D fg1; g7g D C c

s

G4 D fg1; g6g and G04 D fg1; g8g D C d

s

m D 4 W G5 D fg1; g2; g3; g4g D C4

G6 D fg1; g3; g5; g7g D C c2v

G7 D fg1; g3; g6; g8g D C d2v

m D 8 W G8 D fg1; g2; g3; g4; g5; g6; g7; g8g D C4v:

(4.70)

Let us suppose now that the equilibrium state of our molecule is stable underarbitrary infinitesimal displacements of the atoms in the .x; y/ plane. Moreover,we will also assume that this state is isolated in the sense that within a finite sizeneighborhood around it there are no other equilibrium states.

Next, we shall consider planar vibrations, i.e. vibrations of the molecule in theplane of its equilibrium configuration. Let us excite a vibrational regime of ourmolecule by displacing the atoms from their equilibrium positions in a specificmanner. As a result of such displacements, the initial configuration of the moleculewill have a well defined symmetry described by one of the subgroups of the groupG4v listed in (4.70). Indeed, the first configuration in Fig. 4.2 with atoms displacedarbitrarily corresponds to the symmetry group G1 D C1. In this figure, we depictby arrows the atomic displacements and by thick lines the resulting instantaneousconfigurations of the molecule.

Note that it is essential for our analysis that the symmetry of the instanta-neous configuration of the mechanical system be preserved, during the vibrationalmotions. More precisely, if an atomic pattern at time t0 has a symmetry elementg, this element cannot disappear spontaneously at any t > t0. This proposition,which we call the “symmetry preservation theorem”, can be proved rigorously byexamining the Hamiltonian equations of motion. Let us also note, on the other hand,that spontaneous lowering of the symmetry can occur when the dynamical regimeloses its stability leading most frequently to the appearance of another bush of higherdimension. This important phenomenon, which may be regarded as the dynamicalanalogue of a phase transition was discussed in the previous sections of this chapter.

Thus, all possible vibrational regimes of the square molecule can be classifiedaccording to eight subgroups of the group G D C4v. The different configurationsof the vibrating molecule, as depicted in Fig. 4.2, are the following: an arbitraryquadrangle G1 D C1, Fig. 4.2a, a rotating and pulsating square G5 D C4, Fig. 4.2b,a parallelogram G2 D C2, Fig. 4.2c, a rectangle G6 D C c

2v, Fig. 4.2d, a trapezoidG3 D C c

s , Fig. 4.2e, a rhombus G7 D C d2v, Fig. 4.2f, a deltoid G4 D C d

s , Fig. 4.2g,or a pulsating square G8 D C4v, Fig. 4.2h. All these configurations vary in sizeas time progresses, but the type of the corresponding quadrangle does not change.Observe also that these different types of vibrational regimes of the molecule aredescribed by different numbers of dof. Let us discuss this in more detail:

Each of the eight types of vibrational regimes in Fig. 4.2 corresponds to a certainbush of NNMs. For example, the dynamical regime representing a pulsating square


Fig. 4.2 Different vibrational regimes (bushes of NNMs) of the square molecule (after [64])

G D C4v, Fig. 4.2h, can be characterized by only one dof, which can be representedeither by the edge of the square or the displacement of a certain atom from itsequilibrium position along the corresponding diagonal. Thus, such a vibrationalregime is described by a one-dimensional bush consisting only of the so-called“breathing” NNM.

On the other hand, the rhombus-like vibration G7 D C d2v, Fig. 4.2f, and rectangle-

like vibration G6 D C c2v, Fig. 4.2d, are characterized by two dof: The lengths

of the diagonals in the former case and the lengths of the adjacent edges in thelatter. Thus, both of these vibrational regimes are described by two-dimensional


bushes of modes. Similarly, one can check that a trapezoid-type vibration G DC c

s , Fig. 4.2e, corresponds to a three-dimensional bush, the deltoid-type vibrationG4 D C d

s , Fig. 4.2g, to a four-dimensional bush and the vibration with arbitraryquadrangle G1 D C1, Fig. 4.2a, is represented by a five-dimensional bush ofvibrational modes. Note, finally, that vibrations belonging to the G5 D C4 groupof Fig. 4.2b are described by a two-dimensional bush consisting of one rotating andone pulsating mode.

4.5.2 Bushes of NNMs for a Simple Octahedral Molecule

We end this section by describing the occurrence of bushes in a three-dimensionalmechanical system consisting of a molecule with six atoms, whose interactions aredescribed by an isotropic pair-particle potential V.r/ depending only on the distancebetween two particles. Here, we suppose that, at equilibrium, these particles forma regular octahedron with edge a0 as shown in Fig. 4.3. In a Cartesian coordinatesystem four particles of the octahedron lie in the .x; y/ plane and form a squarewith edge a0, while the other two particles are located on the z-axis and are called“top particle” and “bottom particle” with respect to the direction of the z-axis. Thedistance between these particles is also equal to a0.

At equilibrium, this system possesses the point symmetry group Oh, whosepossible bushes of vibrational modes have been listed in [86]. We may suppose,for example, that u.r/ is the well-known Lennard-Jones potential

u.r/ D 1

r12� 1

r6; (4.71)

as was done in [86], where the stability of the bushes in the octahedral moleculewas also studied. In the present discussion, however, we shall consider u.r/ as anarbitrary pair-potential. Let us now write the potential energy of our system in itsvibrational state in the form

Fig. 4.3 Model of anoctahedral moleculeconsisting of six atoms(after [64])


V.X/ DX

i;j.i<j /

u.rij /; (4.72)

where rij is the distance between the i th and j th particles. Here, the N -dimensionalvector X D fx1.t/; x2.t/; : : : ; xN .t/g in (4.72) determines the displacements ofall particles at an arbitrary instant t , while N D 18 is the number of dof of theoctahedral mechanical system.

The configuration vector X.t/ D fx1.t/; x2.t/; : : : ; x18.t/g determines all thedynamical variables xi .t/ as follows: The first three components of this vectorcorrespond to the x, y and z displacements of particle 1, the next three componentscorrespond to x, y, z displacements of particle 2, etc., as numbered in Fig. 4.3. Thedynamics of this system is described by Newton’s equations

Rxi D � @V

@xi

; i D 1; 2; : : : ; 18;

with all particles having mass equal to unity. Each bush of modes correspondsto a certain subgroup Gj of the symmetry group G D Oh of the mechanicalsystem at equilibrium. This means that the vibrational state described by the bushpossesses the symmetry group Gj � G and, therefore, there exist certain restrictionson the dynamical variables xi .t/. As a result, the number m of dof correspondingto the given bush (i.e. its dimension), is smaller than the total number of dynamicalvariables of the system.

Indeed, at any time t , the atom configuration in a vibrational state representsa polyhedron characterized by the symmetry group Gj � Oh. Thus, instead ofthe old variables xi .t/, we introduce m new variables yj .t/ .j D 1; : : : ; m/, whichcompletely determine this polyhedron. Next, we employ the well-known Lagrangemethod (see e.g. [209]) for obtaining the dynamical equations in terms of the newvariables yj.t/. For this purpose, we introduce the Lagrange function L D T � V ,where T is the kinetic and V the potential energy expressed as functions of the newvariables yj and new momenta Pyj .j D 1; : : : ; m/. The Euler-Lagrange equationsin these variables read

d

dt

�@L

@ Pyj

�� @L

@yj

D 0; j D 1; : : : ; m; (4.73)

where m is the dimension of the considered bush. We will not present here thetedious derivation of these equations for the octahedral molecule (see e.g. [86],where dynamical equations for bushes are obtained by an appropriate MAPLEprogram). Rather, we will give the final results for the bushes BŒOh�, BŒD4h�

and BŒC4v�. Here, in the square brackets next to the bush symbol B, we give thesymmetry group of the corresponding bush of vibrational modes. The symmetrygroups of the above bushes satisfy the following group-subgroup relations


C4v � D4h � Oh: (4.74)

The bushes BŒOh�, BŒD4h�, and BŒC4v� have one, two and three dimensions,respectively, while their geometrical features can be immediately identified from thesymmetry groups that correspond to them. Indeed, the one-dimensional bush BŒOh�

consists of only one (“breathing”) mode: The appropriate nonlinear dynamicalregime describes the evolution of a regular octahedron whose edge a D a.t/ period-ically changes in time. The two-dimensional bush BŒD4h� describes a dynamicalregime with two dof. The symmetry group G D D4h of this bush contains thefourfold axis symmetry coinciding with the z coordinate axis and the mirror planecoinciding with the x; y plane. This symmetry group severely restricts the shape ofthe polyhedron describing our mechanical system in the vibrational state. Indeed,the presence of the fourfold axis demands that the quadrangle in the x; y plane bea square. For the same reason, the four edges connecting the particles in the .x; y/

plane at the vertices of the above square with the top particle lying on the z axismust have the same length, denoted here by b.t/.

Similarly, let the length of the edges connecting the bottom particle on the z axiswith any of the four particles in the .x; y/ plane be denoted by c.t/. In the caseof the bush BŒD4h�, b.t/ D c.t/ for any t , due to the horizontal mirror plane in thegroup G D D4h. However, for the three-dimensional bush BŒC4v�, this mirror planeis absent and, therefore, b.t/ ¤ c.t/. In Fig. 4.4 we illustrate the instantaneousconfiguration of our mechanical system vibrating according to the bush BŒC4v�.

Let us also introduce the heights, h1.t/ and h2.t/ for the perpendicular distances,respectively, of the top and bottom vertices of our polyhedron from the .x; y/ plane.We can now write the dynamical equations of the above bushes in terms of thepurely geometrical variables a.t/, b.t/, c.t/, h1.t/ and h2.t/. Choosing a.t/ andh.t/ � h1.t/ � h2.t/ as suitable variables for describing the two-dimensional bushBŒD4h� and a.t/; h1.t/ and h2.t/ as appropriate for describing the three-dimensionalbush BŒC4v�, we express the potential energy for our bushes of vibrational modes asfollows

X

Z

Y

a

b

c

h1

h2

Fig. 4.4 The distortedoctahedron shown hereillustrates the positions of theparticles for the bush BŒC4v�

at a fixed instant in time (after[64])


BŒOh� W V.a/ D 12u.a/ C 3u.p

2a/;

BŒD4h� W V.a; h/ D 4u.a/ C 2u.p

2a/ C 8u

�qh2 C a2

2

�C u.2h/;

BŒC4v� W V.a; h1; h2/ D 4u.a/ C 2u.p

2a/ C 4u.b/ C 4u.c/ C u.h1 C h2/;

where b Dq

a2

2C �

54h1 � 1

4h2

�2, c D

qa2

2C �

54h2 � 1

4h1

�2.

Using then Euler-Lagrange equations, we obtain the following dynamical equa-tions for the above bushes of vibrational modes

BŒOh� WRa D �4u0.a/ � p

2u0.p

2a/IBŒD4h� W

Ra D �2u0.a/ � p2u0.

p2a/ � 2u0.b/ a

b;

Rh D �4u0.b/ hb

� u0.2h/IBŒC4v� W

Ra D �2u0.a/ � p2u0.

p2a/ � u0.b/ a

b� u0.c/ a

c;

Rh1 D �u0.b/ 5h1�h2

b� u0.h1 C h2/;

Rh2 D �u0.c/ 5h2�h1

c� u0.h1 C h2/:

(4.75)

Thus, we have derived dynamical equations for our bushes of vibrational modesin terms of variables having an explicit geometrical meaning. Each bush describesa certain nonlinear dynamical regime corresponding to a vibrational state of theconsidered system, so that at any fixed time the configuration of this system isrepresented by a definite polyhedron characterized by the symmetry group G of thegiven bush. The above dynamical equations for the bushes BŒOh�, BŒD4h� and BŒC4v�

are exact, but rather complicated to solve. One may thus consider approximateequations, which can be obtained by expanding the potential energy in Taylor series(near the equilibrium state) keeping only the lowest order terms in the expansion.Using then the first few leading terms in such decompositions, it turns out thatbushes belonging to systems of different physical nature, with different symmetrygroups and different structures, can produce equivalent dynamical equations. Moreprecisely, the equations for many bushes can be written in the same form, andthis fact leads to the idea of “classes of dynamical universality” mentioned in[81, 82, 294, 295].

Exercises

Exercise 4.1. Write the equations of motion of N coupled harmonic oscillatorsunder fixed boundary conditions (4.2) as a system of ODEs whose rhs is expressedin terms of a tridiagonal matrix S . Using the eigenvectors and eigenvalues of thismatrix, perform a basis transformation that diagonalizes S and change to new


variables Qq , given by (4.3), where the eigenvalues of S , �q D !2q provide the N

normal mode frequencies (4.5). Plot the frequency spectrum !q , q D 1; 2; : : : ; N asa function of q. What can you say about the linear independence of these frequenciesfor general values of N ? Can you find N for which they are linearly dependent?Hint: Consider the cases N C 1 is a prime or a (positive integer) power of 2.

Exercise 4.2. Repeat the analysis of Exercise 4.1 for N coupled harmonic oscil-lators under periodic boundary conditions and derive expressions (4.6) and (4.7).Note that the matrix S is not exactly tridiagonal in this case, as it contains non-zero entries in its S1;N and SN;1 elements. Still it can be diagonalized and and itseigenvalues provide the normal mode frequencies !q in the same manner. Now plotthese frequencies vs. q and comment on their commensurability (linear dependence)properties.

Problems

Problem 4.1. Consider an FPU-ˇ chain with N=12 atoms and periodic boundaryconditions. Study the stability of all six NNMs listed in (4.18)–(4.22) by straight-forward integration of the nonlinear dynamical equations of the FPU-ˇ model fordifferent values of their amplitudes. Check that these NNMs are stable for smallvalues of the initial amplitude A. Find maximal values of A for which the dynamicalregime corresponding to the considered NNM loses its stability. Compare yourresults with those presented in [87] in the form of universal stability diagrams, whichcan be used to analyse the stability of NNMs in the FPU-ˇ chain with an arbitrarynumber of atoms.

Problem 4.2. Consider the plane square molecule discussed in Sect. 4.5.1, whoseatoms interact via the Lennard-Jones potential (4.71). Note that in dimensionlessvariables you may take A D B D 1. Using a mathematical package like MAPLEor MATHEMATICA, find the potential energy of this molecule in the harmonicapproximation and obtain all linear normal modes, as well as their frequencies byfinding eigenvalues and eigenvectors of the matrix (4.45). Show that the squareequilibrium state turns out to be unstable with respect to the rhombus distortion (theeigenfrequency of the mode with rhombus symmetry turns out to be an imaginarynumber!). Hint: additional information can be found in Appendix B of [87], andin [86].

Problem 4.3. Place an additional atom to the center of the square molecule ofSect. 4.5.1, whose interaction with other four atoms is described by Lennard-Jonespotential with coefficients A; B different from those used in Problem 4.2. Showthat one can choose such values of A and B for this potential so that the squareconfiguration of the molecule becomes stable, as all normal mode frequencies turnout to be real. Hint: Consult Appendix B of [81], and [86].


Problem 4.4. Study the nonlinear dynamics of the molecule of Problem 4.3 byintegrating the corresponding dynamical equations, using rhombus linear normalcoordinate as initial displacements of the atoms at the edges of square. Decomposethe configuration vector X.t/ according to (4.42), with 'k being normal coordinatesof the square molecule. Check that only two normal coordinates enter this decom-position: the rhombus-like and square-like (all other linear normal coordinates donot contribute). Thus, you have found one example of a two-dimensional bush. Canyou find in this way another example of a two-dimensional bush of NNMs? Hint:Consult Appendix B of [81].

Problem 4.5. Choosing initial conditions corresponding to other linear normalcoordinates, find all bushes of NNMs for the square molecule depicted in Fig. 4.2.Find also two-dimensional, three-dimensional, etc. bushes for the FPU-ˇ chain.Hint: Consult the results presented in [81] and [82].

Problem 4.6. Following the approach outlined in Sect. 4.5.2 and the referenceslisted therein, show that the analysis of the vibrations of the octahedral moleculedepicted in Figs. 4.3 and 4.4, based on the Euler-Lagrange equations (4.73) leads tothe dynamical equations for the corresponding bushes listed in (4.75). How wouldyou go about solving these equations, if u.r/ is the Lennard-Jones Potential (4.71)?What kind of oscillatory solutions do you find? Hint: Consult results presentedin [86].

Chapter 5Efficient Indicators of Ordered and ChaoticMotion

Abstract This chapter opens with an introduction to the variational equations,derived by the linearization of the ordinary differential equations of a Hamiltoniansystem, and the equations of the tangent map, obtained by linearizing the differenceequations of a symplectic map. We then focus on the particularly efficient strategyof alignment indicators (called SALI and GALI), which employ more than onedeviation vector to uncover the various degrees of chaotic behavior of an orbit,as well as the dimensionality of the torus in the case of quasiperiodic motion.We summarize the derivation of asymptotic formulas of these indicators for longtimes and provide extensive numerical evidence demonstrating their successfulapplication to many examples of Hamiltonian systems and symplectic maps ofinterest to mathematical physics.

5.1 Variational Equations and Tangent Map

As the reader must have undoubtedly realized, one of the principal aims of this bookis to explore in some detail the concepts of local vs. global stability of motion inmulti-dimensional Hamiltonian systems. In particular, we would like to understandthe interrelationship between these concepts and investigate their role in revealingimportant properties of Hamiltonian dynamics in a variety of physical applications.

Global methods, you might rightfully argue, are very rare and quite difficult toapply to systems whose phase space motion varies dramatically from domain todomain. On the other hand, local methods should not be confined only to the analysisof small neighborhoods around fixed points and periodic orbits. As we will find outlater in this book, the existence and stability of certain types of tori of quasiperiodicmotion and the location and size of “strongly” vs. “weakly” chaotic domains are ofcrucial importance to many problems of Hamiltonian dynamics.

Thus, even though we will continue to use local methods to study the flow inthe tangent space of different orbits of Hamiltonian systems, we shall exploit a newapproach that: (1) analyzes as completely as possible the ‘geometrical’ properties of


91

92 5 Efficient Indicators of Ordered and Chaotic Motion

this space, and (2) is rapid and efficient enough to be simultaneously applied to manyinitial conditions, yielding, as fast as possible, a global portrait of the dynamics. Itis this kind of approach that we turn to in the present chapter.

In Chap. 3 we referred to some particular chaos indicators, namely the maximumLyapunov exponent (MLE), the smaller alignment index (SALI) and the generalizedalignment index (GALI), which rely on the evolution of deviation vectors froma given orbit. In this chapter, we will discuss in more detail the behavior of theSALI and the GALI, emphasizing the usefulness of these indices in distinguishingaccurately and efficiently between ordered and chaotic motion.

Let us discuss first, for reasons of consistency, the variational equations govern-ing the evolution of deviation vectors in continuous and discrete time, which wehave already met repeatedly in previous chapters. For this purpose, let us consideran autonomous Hamiltonian system of N dof having a Hamiltonian function

H.q1; q2; : : : ; qN ; p1; p2; : : : ; pN / D h D constant; (5.1)

where qi and pi , i D 1; 2; : : : ; N are the generalized coordinates and conjugatemomenta respectively. An orbit in the 2N -dimensional phase space S of thissystem is defined by the vector

x.t/ D .q1.t/; q2.t/; : : : ; qN .t/; p1.t/; p2.t/; : : : ; pN .t//; (5.2)

with xi D qi , xiCN D pi , i D 1; 2; : : : ; N . The time evolution of this orbit isgoverned by Hamilton’s equations of motion, which in matrix form are given by

Px Dh

@H@p � @H

@q

iT D ˝ � rH; (5.3)

with q D .q1.t/; q2.t/; : : : ; qN .t//, p D .p1.t/; p2.t/; : : : ; pN .t//, and

rH Dh

@H@q1

@H@q2

� � � @H@qN

@H@p1

@H@p2

� � � @H@pN

iT(5.4)

with .T/ denoting the transpose matrix, as we have already seen in Chap. 3. Thematrix ˝ has the following block form

˝ D�

0N IN

�IN 0N

�; (5.5)

(see (3.1) of Sect. 3.1), with IN being the N � N identity matrix and 0N being theN � N matrix with all its elements equal to zero.

An initial deviation vector w.0/ D .ıx1.0/; ıx2.0/; : : : ; ıx2N .0// from an orbitx.t/ evolves in the tangent space TxS of S according to the variational equations

Pw D �˝ � r2H.x.t//

� � w DW B.t/ � w ; (5.6)

5.1 Variational Equations and Tangent Map 93

with r2H.x.t// the Hessian matrix of Hamiltonian (5.1) calculated on the referenceorbit x.t/, i.e.

r2H.x.t//i;j D @2H

@xi @xj

ˇˇx.t/

; i; j D 1; 2; : : : ; 2N: (5.7)

Equations 5.6 are a set of linear differential equations with respect to w, having timedependent coefficients since matrix B.t/ depends on the particular reference orbit,which is a function of time t .

In order to follow the evolution of a deviation vector, the variational equations(5.6) have to be integrated simultaneously with the Hamilton’s equations of motion(5.3), since matrix r2H.t/ depends on the particular reference orbit x.t/, which is asolution of (5.3). Any general purpose numerical integration algorithm can be usedfor the integration of the whole set of (5.3) and (5.6). On the other hand, symplecticintegrators are often preferred when integrating Hamiltonian dynamical systems. Athorough discussion of such methods can be found in [162] (see also [349, 350]).In [312]. So it was shown that it is possible to integrate the Hamilton’s equationsof motion and the corresponding variational equations using the so-called “tangentmap (TM) method”, a scheme based on symplectic integration techniques.

According to this method, a symplectic integrator is used to approximatethe solution of (5.3) by the repeated action of a symplectic map S , while thecorresponding tangent map TS , is used for the integration of (5.6). A simple andsystematic technique to construct TS was also presented in [312]. The TM methodwas proved to be very efficient and superior to other commonly used numericalschemes, both with respect to its accuracy and its speed, for low- [146, 312] andhigh-dimensional Hamiltonian systems [147].

Let us also consider discrete time (t D n 2 N) conservative dynamical systemsdefined by 2N D symplectic map F . A symplectic map is an area-preserving mapwhose Jacobian matrix

M D rF.x/ D @F

@xD

2666664

@F1

@x1

@F1

@x2� � � @F1

@x2N

@F2

@x1

@F2

@x2� � � @F2

@x2N

::::::

:::@F2N

@x1

@F2N

@x2� � � @F2N

@x2N

3777775

; (5.8)

satisfiesM T � ˝ � M D ˝; (5.9)

(see (3.3)). The evolution of an orbit in the 2N -dimensional space S of the map isgoverned by the difference equation

x.n C 1/ � xnC1 D F.xn/: (5.10)


In this case, the evolution of a deviation vector w.n/ � wn, with respect to areference orbit xn, is given by the corresponding tangent map

w.n C 1/ � wnC1 D @F

@x.xn/ � wn: (5.11)

5.2 The SALI Method

The SALI method was introduced in [309] and has been applied successfully todetect regular and chaotic motion in Hamiltonian flows as well as symplecticmaps [55, 57, 72, 233, 234, 242, 245, 262, 269, 319, 320, 322, 325]. The basic ideabehind the success of the SALI method (which essentially distinguishes it fromthe computation of MLE) is the introduction of one additional deviation vectorwith respect to a reference orbit. Indeed, by considering the relation between twodeviation vectors (instead of one deviation vector and the reference orbit), one isable in the case of chaotic orbits to circumvent the difficulty of the slow convergenceof Lyapunov exponents to non-zero values as t ! 1.

As has already been mentioned in Sect. 3.4.1, in order to compute the SALIone follows simultaneously the time evolution of a reference orbit along with twodeviation vectors with initial conditions w1.0/, w2.0/. Since we are only interestedin the directions of these two vectors we normalize them, from time to time, keepingtheir norm equal to 1, setting

Owi .t/ D wi .t/

kwi .t/k ; i D 1; 2; (5.12)

where k � k is the Euclidean norm and the hat (^) over a vector denotes that it is ofunit magnitude. The SALI is then defined as

SALI.t/ D min fk Ow1.t/ C Ow2.t/k ; k Ow1.t/ � Ow2.t/kg ; (5.13)

whence, it is evident that SALI.t/ 2 Œ0;p

2�. SALI D 0 indicates that the twodeviation vectors have become aligned in the same direction (and are equal oropposite to each other); in other words, they are linearly dependent.

In the case of chaotic orbits, the deviation vectors Ow1, Ow2 eventually becomealigned in the direction of MLE, and SALI.t/ falls exponentially to zero. Ananalytical study of SALI’s behavior for chaotic orbits was carried out in [314] whereit was shown that

SALI.t/ / e�.L1�L2/t ; (5.14)

with L1, L2 being the two largest LCEs.In the case of regular motion, on the other hand, the orbit lies on a torus and the

vectors Ow1, Ow2 eventually fall on its tangent space (following a t�1 time evolution)

5.2 The SALI Method 95

having in general different directions as there is no reason for them to becomealigned. This behavior is due to the fact that for regular orbits the norm of a deviationvector increases linearly in time along the flow. Thus, our normalization procedurebrings about a decrease of the magnitude of the coordinates perpendicular to thetorus at a rate proportional to t�1 and so Ow1, Ow2 eventually fall on the tangent spaceof the torus. In this case, the SALI oscillates about values that are different fromzero (for more details see [313]), so that

SALI � const: > 0; t ! 1: (5.15)

A small comment is necessary here. In the case of 2D maps the torus is actuallyan invariant curve and its tangent space is one-dimensional. Thus, in this case, thetwo unit deviation vectors eventually become linearly dependent and SALI becomeszero following a power law, SALI / 1=t2. This is, of course, different than theexponential decay of SALI for chaotic orbits and thus SALI can distinguish easilybetween the two cases even in 2D maps [309]. We note that the analytical derivationof SALI’s behavior for ordered and chaotic orbits will be given in Sect. 5.3.1 below.

Following [309, 314], let us now demonstrate the behavior of the SALI byconsidering some typical ordered and chaotic orbits of low-dimensional dynamicalsystems. Initially we consider an ordered (point A in Fig. 5.1a) and a chaotic orbit(point B in Fig. 5.1a) of the 2D map

x01 D x1 C x2

x02 D x2 � � sin.x1 C x2/

.mod 2�/; (5.16)

for � D 0:5, with initial conditions x1 D 2, x2 D 0 and x1 D 3, x2 D 0, resp-ectively. The evolution of the finite time Lyapunov characteristic number K1

t , havingas limit for t D n ! 1 the MLE (3.42), for both orbits is shown in Fig. 5.1b where

Fig. 5.1 (a) Phase plot of the 2D map (5.16) for � D 0:5. The initial conditions of the orderedorbit A (x1 D 2; x2 D 0) and the chaotic orbit B (x1 D 3; x2 D 0) are marked by black and light-grey filled circles respectively. The evolution of the finite time Lyapunov characteristic number K1

t

(denoted by LN ), and the smaller alignment index SALI, with respect to the number n (denotedby N ) for orbits A (dashed curves) and B (solid curves) is plotted in (b) and (c) respectively(after [309])


it is denoted by LN . K1t of orbit A (dashed line) decreases as the number n of

iterations (which is denoted by N in Fig. 5.1b) increases, following a power law,reaching the value K1

t � 1:6 � 10�6 after 107 iterations. On the other hand, K1t

of the chaotic orbit B (solid line), after some fluctuations seems to stabilize near aconstant value and becomes K1

t � 5 � 10�2 after 107 iterations.As we have already explained, since the map (5.16) is 2D any two deviation

vectors tend to coincide or become opposite, both for ordered and chaotic orbits.In particular, the SALI of the ordered orbit A decreases as n increases, following apower law (SALI / n�2) and it becomes SALI � 10�13 after 107 iterations, whichmeans that the two deviation vectors practically coincide (Fig. 5.1c—dashed curve).On the other hand, the SALI of the chaotic orbit B decreases exponentially fastreaching the limit of accuracy of the computer (10�16) after about 200 iterations(Fig. 5.1c—solid curve). After that time the two vectors are identical since theircoordinates are represented by the same (or opposite) numbers in the computer. So,the SALI can distinguish between ordered and chaotic motion in a 2D map, since itdecreases to zero following completely different time rates.

Let us now study the case of the 4D map

x01 D x1 C x2

x02 D x2 � � sin.x1 C x2/ � �Œ1 � cos.x1 C x2 C x3 C x4/�

x03 D x3 C x4

x04 D x4 � � sin.x3 C x4/ � �Œ1 � cos.x1 C x2 C x3 C x4/�

.mod 2�/; (5.17)

for � D 0:5, � D 0:1 and � D 10�3 by considering the orbits C and D withinitial conditions x1 D 0:5, x2 D 0, x3 D 0:5, x4 D 0 and x1 D 3, x2 D 0,x3 D 0:5, x4 D 0 respectively. The projections on the x1x2 plane of some thousandsof consequents of these orbits, as well as their initial conditions are shown inFig. 5.2a. From this figure we can easily guess that orbit C is ordered, since its

Fig. 5.2 (a) Projections on the x1x2 plane of the orbits with initial conditions x1 D 0:5, x2 D 0,x3 D 0:5, x4 D 0 (ordered orbit C) and x1 D 3, x2 D 0, x3 D 0:5, x4 D 0 (chaotic orbit D) of the4D map (5.17) for � D 0:5, � D 0:1, � D 10�3. The initial conditions of the orbits are markedby filled circles. The evolution of K1

t (denoted by LN ), and the SALI, with respect to the numbern of iterations (denoted by N ), for the ordered orbit C (dashed curves) and for the chaotic orbit D(solid curves) is plotted in (b) and (c) respectively (after [309])


projection on the x1x2 plane forms a closed curve, while orbit D is chaotic, since itproduces scattered points on the x1x2 plane. This guess is verified from the evolutionof K1

t (denoted by LN ), shown in Fig. 5.2b.In Fig. 5.2c we show the evolution of the SALI for these two orbits. For the

ordered orbit C the SALI remains almost constant, fluctuating around SALI � 0:28,while it decreases abruptly in the case of the chaotic orbit D, reaching the limit ofaccuracy of the computer (10�16) after about 4:7�103 iterations. After that time thecoordinates of the two vectors are represented by opposite numbers in the computer,and the chaotic nature of the orbit is determined beyond any doubt. On the otherhand, the computation of the MLE (Fig. 5.2b) is not able to definitely characterizethe orbit as chaotic after the same number of iterations, since it is not yet clear if K1

t

will tend to a positive value.Let us now verify the validity of (5.14) for two simple Hamiltonian systems

with two and three dof: the well-known Henon-Heiles system [164], having theHamiltonian function (2.36)

H2 D 1

2.p2

x C p2y/ C 1

2.x2 C y2/ C x2y � 1

3y3; (5.18)

and the three dof Hamiltonian

H3 D 1

2.p2

x C p2y C p2

z / C 1

2.Ax2 C By2 C C z2/ � �xz2 � �yz2; (5.19)

studied in [100,101]. We keep the parameters of the two systems fixed at the energiesH2 D 0:125 and H3 D 0:00765, with A D 0:9, B D 0:4, C D 0:225, � D 0:56 and� D 0:2.

We recall that two dof Hamiltonian systems have only one positive LCE, L1,since the second largest is L2 D 0. So, (5.14) becomes for such systems

SALI.t/ / e�L1t : (5.20)

In Fig. 5.3a we plot in log-log scale K1t (denoted by Lt ) as a function of time t

for a chaotic orbit of the system (5.18). K1t remains different from zero, which

implies the chaotic nature of the orbit. Following its evolution for a sufficiently longtime interval to achieve reliable estimates (t � 10; 000) we obtain L1 � 0:047. InFig. 5.3b we plot the SALI for the same orbit (solid curve) using linear scale for thetime t . Again we conclude that the orbit is chaotic as SALI � 10�16 for t � 800. If(5.20) is valid, the slope of the SALI in Fig. 5.3b should be given approximately by�L1= ln 10, because log.SALI/ is a linear function of t . From Fig. 5.3b we see thata line having precisely this slope with L1 D 0:047 (dashed line) approximate quiteaccurately the computed values of the SALI. We note here that a MAPLE algorithmfor the computation of the SALI for orbits of the Henon-Heiles system is presentedin Appendix B.


a b

Fig. 5.3 (a) The evolution of K1t (denoted by Lt ) for the chaotic orbit with initial condition x D 0,

y D �0:25, px ' 0:42081, py D 0 of the 2D system (5.18). (b) The SALI of the same orbit(solid curve) and a function proportional to e�L1t (dashed curve) for L1 D 0:047. Note that thet-axis is linear (after [309])

a b

Fig. 5.4 (a) The evolution of the two finite time Lyapunov characteristic numbers K1t , K2

t

(denoted by 1t and 2t respectively) for the chaotic orbit with initial condition x D 0, y D 0,z D 0, px D 0, py D 0, pz ' 0:123693 of the 3D system (5.19). (b) The SALI of the sameorbit (solid curve) and a function proportional to e�.L1�L2/t (dashed curve) for L1 D 0:0107,L2 D 0:0005. Note that the t-axis is linear (after [309])

Chaotic orbits of Hamiltonian systems with three dof generally have two positiveLyapunov exponents, L1 and L2. So, for approximating the behavior of the SALIby (5.14), both L1 and L2 are needed. We compute L1, L2 for a chaotic orbit of the3D system (5.19) as the long time estimates of some appropriate quantities, K1

t , K2t ,

by applying the method proposed by Benettin et al. [32] (see also [310] for moredetails). The results are presented in Fig. 5.4a. The computation is carried out untilK1

t and K2t stop having high fluctuations and approach some non-zero values (since

the orbit is chaotic), which could be considered as good approximations of their


Fig. 5.5 (a) The x D 0 PSS of the 2D Henon-Heiles system (5.18). The axis py D 0 is alsoplotted. (b) The values of the SALI at t D 1; 000 for orbits with initial conditions on the py D 0

line of panel (a) as a function of the y coordinate of the initial condition. (c) Regions of differentvalues of the SALI on the PSS of panel (a) at t D 1; 000. The initial conditions are colored blackif their SALI 6 10�12, deep gray if 10�12 < SALI 6 10�8, gray if 10�8 < SALI 6 10�4 andlight gray if SALI > 10�4 (after [309])

limits L1, L2. Actually for t � 105 we have K1t � 0:0107, K2

t � 0:0005. Usingthese values as good approximations of L1, L2 we see in Fig. 5.4b that the slope ofthe SALI (solid curve) is well reproduced by (5.14) (dashed curve).

Exploiting the advantages of the SALI we can use it to efficiently identify orderedand chaotic orbits in large regions of phase space where large scale ordered andchaotic regions are both present. As an example, we consider the two dof system(5.18), whose PSS for x D 0 is presented in Fig. 5.5a. Regions of ordered motion,around stable periodic orbits, are seen to coexist with chaotic regions filled byscattered points. In order to demonstrate the effectiveness of the SALI method, wefirst consider orbits whose initial conditions lie on the line py D 0. In particularwe take 5; 000 equally spaced initial conditions on this line and compute the valueof the SALI for each one. The results are presented in Fig. 5.5b where we plot theSALI as a function of the y coordinate of the initial condition of these orbits fort D 1; 000. The data points are line connected, so that the changes of the SALIvalues are clearly visible.

Note that there are intervals where the SALI has large values (e.g. larger than10�4), which correspond to ordered motion in the island of stability crossed by thepy D 0 line in Fig. 5.5a. There also exist regions where the SALI has very smallvalues (e.g. smaller than 10�12) denoting that in these regions the motion is chaotic.These intervals correspond to the regions of scattered points crossed by the py D 0

line in Fig. 5.5a. Although most of the initial conditions give large (>10�4) or verysmall (610�12) values for the SALI, there also exist initial conditions that haveintermediate values of the SALI (10�12 < SALI 6 10�4). These initial conditionscorrespond to sticky chaotic orbits, remaining for long time intervals at the bordersof islands, whose chaotic nature will be revealed later on (see also Chap. 8). Notethat it is not easy to define a threshold value, such that if the SALI is smaller, this


reliably signifies chaoticity. Nevertheless, numerical experiments in several systemsshow that in general a good guess for this value could be .10�4.

In Fig. 5.5b, around y � �0:1 there exists a group of points inside a bigchaotic region having SALI > 10�4. These points correspond to orbits with initialconditions inside a small stability island, which is not easily visible in Fig. 5.5a. Alsothe point with y D �0:2088 has very high value of the SALI (>0:1) in Fig. 5.5b,while all its neighboring points have SALI < 10�9. This point actually correspondsto an ordered orbit inside a tiny island of stability, which can be revealed onlyafter a very high magnification of this region of the PSS. Thus, we conclude thatthe systematic application of the SALI method can reveal very fine details of thedynamics.

By carrying out the above analysis for points not only along a line but on thewhole plane of the PSS, and giving to each point a color according to the valueof the SALI, we can have a clear picture of the regions where chaotic or orderedmotion occurs. The outcome of this procedure is presented in Fig. 5.5c where thevalues of the logarithm of the SALI are divided in four intervals. Initial conditionshaving different values of the SALI at t D 1; 000 are plotted by different shadesof gray: black if SALI 6 10�12, deep gray if 10�12 < SALI 6 10�8, gray if10�8 < SALI 6 10�4 and light gray if SALI > 10�4. Thus, in Fig. 5.5c weclearly distinguish between light gray regions, where the motion is ordered andblack regions, where it is chaotic. At the borders between these regions we finddeep gray and gray points, which correspond to sticky chaotic orbits. It is worth-mentioning that in Fig. 5.5c we can see small islands of stability inside the largechaotic sea, which are not visible in the PSS of Fig. 5.5a, like the one for y � �0:1,py � 0.

As an example of SALI’s usefulness for investigating dynamical systems wepresent the application of the index to a model of a simplified accelerator ring withlinear frequencies (tunes) qx , qy , having a localized thin sextupole magnet. Theevolution of a charged particle in this ring is modeled by the 4D symplectic map

0BB@

x01

x02

x03

x04

1CCA D

0BB@

cos !1 � sin !1 0 0

sin !1 cos !1 0 0

0 0 cos !2 � sin !2

0 0 sin !2 cos !2

1CCA �

0BB@

x1

x2 C x21 � x2

3

x3

x4 � 2x1x3

1CCA ; (5.21)

where x1 (x3) denotes the initial deflection from the ideal circular orbit in thehorizontal (vertical) direction before the particle enters the magnetic element, andx2 (x4) is the associated momentum [57,59]. Primes denote positions and momentaafter one turn in the ring. The parameters !1 and !2 are related to the accelerator’stunes qx and qy by !1 D 2�qx and !2 D 2�qy . In [57] the particular case ofqx D 0:61803, qy D 0:4152 was considered and the SALI method was appliedto construct phase space “charts”, where regions of chaos and order were clearly


x1

x 3

−1 −0.5 0 0.5 1

−1

−0.5

a b

0

0.5

1

x1

x 3

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1−8

−6

−4

−2

0

Fig. 5.6 Regions of different SALI values on (a) the .x1; x3/ plane of the uncontrolled map (5.21),and (b) the controlled map constructed in [49]. 16; 000 uniformly distributed initial conditions inthe square .x1; x3/ 2 Œ�1; 1� � Œ�1; 1�, x2.0/ D x4.0/ D 0 are followed for 105 iterations, andthey are colored according to their final log.SALI/ value. The white colored regions correspond toorbits that escape in less than 105 iterations (after [50])

0 0.5 10

25

50

75

100

r

% o

f orb

its

Fig. 5.7 The percentages ofregular (solid curves) andchaotic (dashed curves) orbitsafter n D 105 iterations ofthe uncontrolled map(5.21) (black curves), thecontrolled map of [49] (bluecurves), in a four-dimensionalhypersphere centered atx1 D x2 D x3 D x4 D 0, asa function of the hypersphereradius r (after [50])

identified in Fig. 5.6, and for evaluating the percentages of ordered and chaotic orbitsin a hypersphere centered at the origin x1 D x2 D x3 D x4 D 0 (black curves inFig. 5.7). Later on, in [49, 50] chaos control techniques were used to increase theso-called dynamic aperture (i.e. the stability domain around the ideal circular orbit)of the system. The procedure was successful since the constructed controlled maphas a larger stability region as can be seen from Figs. 5.6 and 5.7.

From all the results presented above we conclude that the SALI offers indeedan easy and efficient method for distinguishing the chaotic vs. ordered nature oforbits in a variety of problems, since it tends exponentially to zero in the case ofchaotic orbits, while it fluctuates around non-zero values for regular trajectoriesof Hamiltonian systems and 2N D symplectic maps with N > 1. In the case of2D maps, the SALI tends to zero both for regular and chaotic orbits but with verydifferent time rates, which allows us again to distinguish between the two cases


[309]. In particular the SALI tends to zero following an exponential law for chaoticorbits and decays to zero following a power law for regular orbits.

Thus, although the behavior of SALI in 2D maps is clearly understood, the factremains that SALI does not always have the same behavior for regular orbits, as itmay oscillate about a constant or decay to zero by a power law, depending on thedimensionality of the tangent space of the reference orbit. It is therefore interestingto ask whether this index can be generalized, so that different power laws may befound to characterize regular motion in higher dimensions.

Let us make one more remark concerning the behavior of SALI for chaotic orbits.Looking at (5.14), one might wonder what would happen in the case of a chaoticorbit whose two largest Lyapunov exponents L1 and L2 are equal or almost equal.Although this may not be common in generic Hamiltonian systems, such cases canbe found in the literature. In one such example presented in [16], very close to aparticular unstable periodic orbit of a 15 dof Hamiltonian system, the two largestLyapunov exponents are nearly equal L1 � L2 � 0:0002. Even though, in thatexample, SALI still tends to zero at the rate indicated by (5.14), it is evident that thechaotic nature of an orbit cannot be revealed very fast by the SALI method. Howcan we overcome this problem? Can we define, for example, a generalized index thatdepends on several Lyapunov exponents (instead of only the two largest ones), asthis might accelerate considerably the identification of chaotic motion? The answeris yes. Such a generalized index exists, and is discussed in detail in the next section.

5.3 The GALI Method

Observe first that seeking the minimum of the two positive quantities in (5.13)(which are bounded above by 2) is essentially equivalent to evaluating the product

P.t/ D k Ow1.t/ C Ow2.t/k � k Ow1.t/ � Ow2.t/k ; (5.22)

at every value of t . Indeed, if the minimum of these two quantities is zero (as inthe case of a chaotic reference orbit), so will be the value of P.t/. On the otherhand, if it is not zero, P.t/ will be proportional to the constant about which thisminimum oscillates (as in the case of regular motion). This suggests that, insteadof computing the SALI.t/ from (5.13), one might as well evaluate the “exterior” or“wedge” product of the two deviation vectors Ow1 ^ Ow2 for which it holds

k Ow1 ^ Ow2k D k Ow1 � Ow2k � k Ow1 C Ow2k2

; (5.23)

and which represents the “area” of the parallelogram formed by the two deviationvectors (see Exercise 5.1). For the definition of the wedge product see the Appendixat the end of this chapter. Indeed, the ‘wedge’ product of two vectors can begeneralized to represent the “volume” of a parallelepiped formed by the deviation

5.3 The GALI Method 103

vectors Ow1; Ow2; : : : ; Owk , 2 � k � 2N , an N dof Hamiltonian system, or a 2N Dsymplectic map.

Hence, the SALI is suitable for checking whether or not two normalizeddeviation vectors Ow1, Ow2 (having norm 1), eventually become linearly dependent, byfalling in the same direction. The linear dependence of the two vectors is equivalentto the vanishing of the “area” of the parallelogram having as edges the two vectors.Generalizing this idea we now follow the evolution of k deviation vectors Ow1, Ow2,: : :, Owk , with 2 � k � 2N , and determine whether these eventually become linearlydependent, by checking if the “volume” of the parallelepiped having these vectorsas edges goes to zero. This volume is equal to the norm of the wedge product ofthese vectors and is defined as the Generalized Alignment Index (GALI) of order k

GALIk.t/ D k Ow1.t/ ^ Ow2.t/ ^ � � � ^ Owk.t/k : (5.24)

Let us see how one can actually evaluate this norm and consequently the GALIk.All normalized deviation vectors Owi , i D 1; 2; : : : ; k, belong to the 2N -dimensionaltangent space of the Hamiltonian flow. Using as a basis of this space the usual set oforthonormal vectors

Oe1 D .1; 0; 0; : : : ; 0/; Oe2 D .0; 1; 0; : : : ; 0/; : : : ; Oe2N D .0; 0; 0; : : : ; 1/; (5.25)

any deviation vector Owi can be written as a linear combination

Owi D2NX

j D1

wij Oej ; i D 1; 2; : : : ; k; (5.26)

where wij are real numbers satisfyingP2N

j D1 w2ij D 1: Thus, (5.26) gives

26664

Ow1

Ow2

:::

Owk

37775 D

26664

w11 w12 � � � w1 2N

w21 w22 � � � w2 2N

::::::

:::

wk1 wk2 � � � wk 2N

37775 �

26664

Oe1

Oe2

:::

Oe2N

37775 D W �

26664

Oe1

Oe2

:::

Oe2N

37775 ; (5.27)

and according to (5.24) the GALIk is evaluated as

GALIk D k Ow1 ^ Ow2 ^ � � � ^ Owkk D

8ˆ<ˆ:

X1�i1<i2<��<ik�2N

ˇˇˇˇˇ

w1i1 w1i2 � � � w1ik

w2i1 w2i2 � � � w2ik:::

::::::

wki1 wki2 � � � wkik

ˇˇˇˇˇ

29>>>>=>>>>;

1=2

;

(5.28)where the sum is performed over all possible combinations of k indices out of 2N .


In order to compute GALIk , therefore, we follow the evolution of an orbit andof k initially linearly independent unit deviation vectors Owi , i D 1; 2; : : : ; k. Fromtime to time we normalize these deviation vectors to unity, keeping their directionsintact, and compute GALIk as the norm of their wedge product using (5.28).

Consequently, if GALIk.t/ tends to zero, this would imply that the volume ofthe parallelepiped having the vectors Owi as edges also shrinks to zero, as at leastone of the deviation vectors becomes linearly dependent on the remaining ones.On the other hand, if GALIk.t/ remains far from zero, as t grows arbitrarily, thiswould indicate the linear independence of the deviation vectors and the existence ofa corresponding parallelepiped, whose volume is different from zero for all time.

Equation 5.28 is ideal for the theoretical determination of the asymptotic behav-ior of GALIs for chaotic and ordered orbits (see Sect. 5.3.1 below). However, froma practical point of view using (5.28) for the numerical evaluation of GALIk is notvery efficient as it might require the computation of a large number of determinants.In [14, 316] a more efficient numerical technique for the computation of GALIk,was proposed. This technique is based on the Singular Value Decomposition of thematrix W in (5.27), whose rows are the coordinates of the unit deviation vectors Owi ,i D 1; 2; : : : ; k. In particular, it has been shown that GALIk is equal to the productof the singular values zi � 0, i D 1; 2; : : : ; k of W T

GALIk DkY

iD1

zi ; (5.29)

(see Exercise 5.2).

5.3.1 Theoretical Results for the Time Evolution of GALI

5.3.1.1 Exponential Decay of GALI for Chaotic Orbits

In order to investigate the dynamics in the vicinity of a chaotic orbit of an N dofHamiltonian system or a 2N D symplectic map, let us first recall some well-knownproperties of the LCEs of these systems, following for example [310]. Oseledec[259] has shown that the mean exponential rate of divergence L .x.0/; w/ from areference orbit with initial condition x.0/ given by

L .x.0/; w/ D limt!1

1

tln

kw.t/kkw.0/k ; (5.30)

exists and is finite. Furthermore, there is a 2N -dimensional basis fOu1; Ou2; : : : ; Ou2N gof the system’s tangent space so that L .x.0/; w/ takes one of the 2N (possibly non-distinct) values

Li .x.0// D L .x.0/; Oui / ; i D 1; 2; : : : ; 2N (5.31)


which are the LCEs ordered in size as

L1 � L2 � : : : � L2N : (5.32)

In addition, it has been proven in [32] that the LCEs are grouped in pairs of oppositesign, i.e. Li D �L2N �iC1, i D 1; 2; : : : ; N , while two of them are equal to zero(LN D LN C1 D 0) in the case of autonomous Hamiltonian systems.

The time evolution of an initial deviation vector

w.0/ D2NXiD1

ci Oui ; (5.33)

can be approximated by

w.t/ D2NXiD1

ci edi t Oui ; (5.34)

where ci , di are real numbers depending on the specific phase space location throughwhich the reference orbit passes. Thus, the quantities di , i D 1; 2; : : : ; 2N may bethought of as “local Lyapunov exponents” having as limits for t ! 1 the LCEs Li ,i D 1; 2; : : : ; 2N .

Assuming that, after a certain time interval, the di , i D 1; 2; : : : ; 2N , do notfluctuate significantly about their limiting values, we write di � Li and express theevolution of the deviation vectors wi in the form

wi .t/ D2NX

j D1

cij eLj t Ouj : (5.35)

Thus, if L1 > L2, a leading order estimate of the deviation vector’s Euclidean norm(for t large enough), is given by

kwi .t/k � jci1jeL1t : (5.36)

Consequently, the matrix W in (5.27) of coefficients of k normalized deviationvectors Owi .t/ D wi .t/=kwi .t/k, i D 1; 2; : : : ; k, with 2 � k � 2N , using asbasis of the vector space the set fOu1; Ou2; : : : ; Ou2N g becomes

C.t/ D

266666664

s1c1

2

jc11 j e

�.L1�L2/t c13

jc11 je

�.L1�L3/t � � � c12N

jc11 j e

�.L1�L2N /t

s2c2

2

jc21 j e

�.L1�L2/t c23

jc21 je

�.L1�L3/t � � � c22N

jc21 j e

�.L1�L2N /t

::::::

::::::

skck

2

jck1 je

�.L1�L2/t ck3

jck1 je

�.L1�L3/t � � � ck2N

jck1 je

�.L1�L2N /t

377777775

; (5.37)

with si D sign.ci1/ and i D 1; 2; : : : ; k.


Note that this matrix is not identical to the one appearing in (5.28) as it is notexpressed with respect to the basis (5.25). Nevertheless, we shall proceed with ouranalysis using matrix C.t/, as we do not expect that the use of a different basisaffects significantly our results since the sets fOuig and f Oeig, i D 1; 2; : : : ; 2N , arevalid bases of the vector space, related by a non-singular transformation of the form� Ou1 Ou2 : : : Ou2N

�T D Tc � � Oe1 Oe2 : : : Oe2N

�T, with Tc denoting the transformation

matrix. So, by studying analytically the time evolution of the determinants of matrixC.t/, we expect to derive accurate approximations of the behavior of the GALIk

(5.24) for chaotic orbits. The validity of this approximation is numerically verifiedin Sect. 5.3.2.

From all the possible k � k determinants of C.t/ (5.37), the one that decreasesthe slowest is

D1;2;3;:::;k D

ˇˇˇˇˇˇˇ

s1c1

2

jc11 je

�.L1�L2/t � � � c1jk

jc11 je

�.L1�Ljk/t

s2c2

2

jc21 je

�.L1�L2/t � � � c2jk

jc21 je

�.L1�Ljk/t

::::::

:::

skck

2

jck1 je

�.L1�L2/t � � � ckjk

jck1 je

�.L1�Ljk/t

ˇˇˇˇˇˇˇ

D

D

ˇˇˇˇˇˇˇ

s1c1

2

jc11 j � � � c1

jk

jc11 j

s2c2

2

jc21 j � � � c2

jk

jc21 j

::::::

:::

skck

2

jck1 j � � � ck

jk

jck1 j �

ˇˇˇˇˇˇˇ

e�Œ.L1�L2/C.L1�L3/C��C.L1�Ljk/�t )

D1;2;3;:::;k / e�Œ.L1�L2/C.L1�L3/C��C.L1�Lk/�t : (5.38)

All other determinants of C.t/ tend to zero faster than D1;2;3;:::;k and we, therefore,conclude that the rate of decrease of GALIk is dominated by (5.38), yielding theapproximation

GALIk.t/ / e�Œ.L1�L2/C.L1�L3/C��C.L1�Lk /�t : (5.39)

In the previous analysis we assumed that L1 > L2 so that the norm of eachdeviation vector can be well approximated by (5.36). If the first m LCEs, with 1 <

m < k, are equal, or very close to each other, i.e. L1 ' L2 ' � � � ' Lm, (5.39)becomes

GALIk.t/ / e�Œ.L1�LmC1/C.L1�LmC2/C��C.L1�Lk/�t ; (5.40)

which still describes an exponential decay. However, for k � m < N the GALIk

does not tend to zero as there exists at least one determinant of C.t/ that doesnot vanish. In this case, of course, one should increase the number of deviationvectors until an exponential decrease of GALIk is achieved. The extreme situation


that all Li D 0 corresponds to motion on quasiperiodic tori, where all orbits areregular and is described below.

5.3.1.2 The Evaluation of GALI for Ordered Orbits

Ordered orbits of an N dof Hamiltonian system (or a 2N D symplectic map)typically lie on N -dimensional tori found around stable periodic orbits. Such torican be accurately described by N formal local integrals of motion in involution, sothat the system appears locally integrable. This means that we could perform a localtransformation to action-angle variables, considering as actions J1; J2; : : : ; JN thevalues of the N formal integrals, so that Hamilton’s equations of motion, locallyattain the form PJi D 0

Pi D !i .J1; J2; : : : ; JN /i D 1; 2; : : : ; N: (5.41)

These can be easily integrated to give

Ji .t/ D Ji0

i .t/ D i0 C !i .J10; J20; : : : ; JN 0/ ti D 1; 2; : : : ; N; (5.42)

where Ji0, i0, i D 1; 2; : : : ; N are the initial conditions.By denoting as �i , �i , i D 1; 2; : : : ; N , small deviations of Ji and i respectively,

the variational equations of system (5.41), describing the evolution of a deviationvector are P�i D 0

P�i D PNj D1 !ij �j

i D 1; 2; : : : ; N; (5.43)

where

!ij D @!i

@Jj

jJ0 i; j D 1; 2; : : : ; N; (5.44)

and J0 D .J10; J20; : : : ; JN 0/ D constant, represents the N –dimensional vector ofthe initial actions. The solution of these equations is:

�i .t/ D �i .0/

�i .t/ D �i .0/ ChPN

j D1 !ij �j .0/i

ti D 1; 2; : : : ; N: (5.45)

From (5.45) we see that an initial deviation vector w.0/ with coordinates �i .0/,i D 1; 2; : : : ; N , in the action variables and �i .0/, i D 1; 2; : : : ; N in the angles,i.e. w.0/ D .�1.0/; �2.0/; : : : ; �N .0/; �1.0/; �2.0/; : : : ; �N .0//, evolves in time insuch a way that its action coordinates remain constant, while its angle coordinatesincrease linearly in time. This behavior implies an almost linear increase of the normof the deviation vector.

Using as a basis of the 2N -dimensional tangent space of the Hamiltonian flowthe 2N unit vectors fOv1; Ov2; : : : ; Ov2N g, such that the first N of them, Ov1; Ov2; : : : ; OvN ,


correspond to the N action variables and the remaining ones, OvN C1; OvN C2; : : : ; Ov2N ,to the N conjugate angle variables, any unit deviation vector Owi , i D 1; 2; : : :, canbe written as

Owi .t/ D 1

kw.t/k

24

NXj D1

�ij .0/ Ovj C

NXj D1

�i

j .0/ CNX

kD1

!kj �ij .0/t

!OvN Cj

35: (5.46)

We point out that the quantities !ij , i; j D 1; 2 : : : ; N , in (5.44), depend only on theparticular reference orbit and not on the choice of the deviation vector. We also notethat the basis Ovi , i D 1; 2; : : : ; 2N depends on the specific torus on which the motionoccurs and is related to the usual vector basis Oei , i D 1; 2; : : : ; 2N , (5.25), througha non-singular transformation. The basis f Oe1; Oe2; : : : ; Oe2N g is used to describe theevolution of a deviation vector with respect to the original qi , pi i D 1; 2; : : : ; N

coordinates of the Hamiltonian system, while the basis fOv1; Ov2; : : : ; Ov2N g is used todescribe the same evolution in action-angle variables, so that the equations of motionare the ones given by (5.41).

Let us now study the case of k, general, linearly independent unit deviationvectors f Ow1; Ow2; : : : ; Owkg with 2 � k � 2N . As we have already explained, thek deviation vectors will eventually fall on the N -dimensional tangent space of thetorus on which the motion occurs. In their final state, the deviation vectors will havecoordinates only in the N –dimensional space spanned by OvN C1; OvN C2; : : : ; Ov2N .Now, if we start with 2 � k � N general deviation vectors there is no particularreason for them to become linearly dependent and their wedge product will bedifferent from zero, yielding GALIk which are not zero. However, if we start withN < k � 2N deviation vectors, some of them will necessarily become linearlydependent. Thus, in this case, their wedge product, as well as GALIk will be zero.

Let us examine in more detail the behavior of GALIs in these cases. The matrixD.t/ having as rows the coordinates of the deviation vectors with respect to thebasis fOv1; Ov2; : : : ; Ov2N g has the form

D.t/ D 1QkmD1 kwm.t/k

�

26664

�11 .0/ � � � �1

N .0/ �11.0/ CPN

mD1 !1m�1m.0/t � � � �1

N .0/ CPNmD1 !Nm�1

m.0/t

�21 .0/ � � � �2

N .0/ �21.0/ CPN

mD1 !1m�2m.0/t � � � �2

N .0/ CPNmD1 !Nm�2

m.0/t:::

::::::

:::

�k1 .0/ � � � �k

N .0/ �k1.0/ CPN

mD1 !1m�km.0/t � � � �k

N .0/ CPNmD1 !Nm�k

m.0/t

37775;

(5.47)

where i D 1; 2; : : : ; k. Denoting by �ki and �k

i the k � 1 column matrices

�ki D �

�1i .0/ �2

i .0/ : : : �ki .0/

�T; �k

i D ��1

i .0/ �2i .0/ : : : �k

i .0/�T

; (5.48)


the matrix D.t/ of (5.47) assumes the much simpler form

D.t/ D 1QkiD1 Mi.t/

� ��k1 : : : �k

N �k1 CPN

iD1 !1i �ki t : : : �k

N CPNiD1 !N i �

ki t�

D 1QkiD1 Mi.t/

� Dk.t/: (5.49)

So all determinants appearing in the definition of GALIk have as a commonfactor the quantity 1=

QkiD1 Mi.t/. This quantity decreases to zero according to the

power law1Qk

iD1 Mi.t// 1

tk; (5.50)

due to the fact that the norm of each deviation vector grows linearly with time t

for long times. In order to determine the asymptotic time evolution of GALIk , wesearch for the fastest increasing determinants of all the possible k � k minors of thematrix Dk in (5.49), as time t grows.

Let us start with k being less than or equal to the dimension of the tangent spaceof the torus, i.e. 2 � k � N . The fastest increasing determinants in this case arethe N Š=.kŠ.N � k/Š/ determinants, whose k columns are chosen among the last N

columns of matrix Dk :

�kj1;j2;:::;jk

Dˇˇ�k

j1CPN

iD1 !j1i �ki t �k

j2CPN

iD1 !j2i �ki t � � � �k

jkCPN

iD1 !jki�ki t

ˇˇ ;

(5.51)with 1 � j1 < j2 < : : : < jk � N . Using standard properties of determinants, weeasily see that the time evolution of �k

j1;j2;:::;jkis mainly determined by the behavior

of determinants of the form

ˇ!j1m1�

km1

t !j2m2�km2

t � � � !jkmk�k

mktˇ D tk

kYiD1

!ji mi � ˇ �km1

�km2

� � � �kmk

ˇ / tk ;

(5.52)where mi 2 f1; 2; : : : ; N g, i D 1; 2; : : : ; k, with mi ¤ mj , for all i ¤ j . Thus,from (5.50) to (5.52) we conclude that the contribution to the behavior of GALIk

of the determinants related to �kj1;j2;:::;jk

is to provide constant terms in (5.28). Allother determinants appearing in the definition of GALIk , not being of the form of�k

j1;j2;:::;jk, contain at least one column from the first N columns of matrix Dk and

introduce in (5.28) terms that grow at a rate slower than tk , which will ultimatelyhave no bearing on the behavior of GALIk(t). Thus, we get

GALIk.t/ � constant for 2 � k � N: (5.53)

Next, we turn to the case of k deviation vectors with N < k � 2N . The fastestgrowing determinants are again those containing the last N columns of the matrixDk , i.e.


�kj1;j2;:::;jk�N ;1;2;:::;N D

ˇˇ�k

j1� � � �k

jk�N�k

1 CPNiD1 !1i �

ki t � � � �k

N CPNiD1 !N i �

ki t

ˇˇ;

(5.54)with 1 � j1 < j2 < : : : < jk�N � N . The first k � N columns of�k

j1;j2;:::;jk�N ;1;2;:::;N are chosen among the first N columns of Dk which are timeindependent. So there exist N Š=..k�N /Š.2N �k/Š/ determinants of the form (5.54),which can be written as a sum of simpler k � k determinants, each containing in theposition of its last N columns �k

i , i D 1; 2; : : : ; N and/or columns of the form!j i�

ki t with i; j D 1; 2; : : : ; N . We exclude the ones where �k

i , i D 1; 2; : : : ; N

appear more than once, since in that case the corresponding determinant is zero.Among the remaining determinants, the fastest increasing ones are those containingas many columns proportional to t as possible.

Since t is always multiplied by the �ki , and such columns occupy the first k � N

columns of �kj1;j2;:::;jk�N ;1;2;:::;N , t appears at most N � .k � N / D 2N � k times.

Otherwise the determinant would contain the same �ki column at least twice and

would be equal to zero. The remaining k � .2N � k/ � .k � N / D k � N columnsare filled by the �k

i each of which appears at most once. Thus, the time evolution of�k

j1;j2;:::;jk�N ;1;2;:::;N is mainly expressed by determinants of the form

ˇˇ�k

j1� � � �k

jk�N�k

i1� � � �k

ik�N!ik�N C1m1�

km1

t � � � !iN m2N �k�k

i2N �kt

ˇˇ / t2N �k;

(5.55)with il 2 f1; 2; : : : ; N g, l D 1; 2; : : : ; N , il ¤ ij , for all l ¤ j and ml 2f1; 2; : : : ; N g, l D 1; 2; : : : ; 2N � k, ml 62 fj1; j2; : : : ; jk�N g, ml ¤ mj , for alll ¤ j . So determinants of the form (5.54) contribute to the time evolution of GALIk

by introducing terms proportional to t2N �k=tk D 1=t2.k�N /. All other determinantsappearing in the definition of GALIk, not having the form of �k

j1;j2;:::;jk�N ;1;2;:::;N ,

introduce terms that tend to zero faster than 1=t2.k�N / since they contain more thank � N time independent columns of the form �k

i , i D 1; 2; : : : ; N . Thus, GALIk

tends to zero following a power law of the form

GALIk.t/ / 1

t2.k�N /for N < k � 2N: (5.56)

In summary, the behavior of GALIk for ordered orbits lying on an N -dimensionaltorus in the 2N -dimensional phase space of an N dof Hamiltonian system (or a2N D symplectic map) is given by

GALIk.t/ /(

constant if 2 � k � N1

t2.k�N / if N < k � 2N: (5.57)

We remark that SALI is practically equivalent to GALI2 as we see from thedefinitions of SALI (5.13) and GALI (5.24), as well as (5.23). We also note thatestimates (5.57) are valid only when the particular conditions of each case are


satisfied. For example, in the case of 2D maps, where the only possible torus isan one-dimensional invariant curve, the tangent space is one-dimensional. Thus, thebehavior of GALI2 (which is the only possible index in this case) is given by thesecond branch of (5.57), i.e. GALI2 / 1=t2, since the first case of (5.57) is notapplicable. The SALI / t�2 behavior has actually been seen in Fig. 5.1c for anordered orbit of the 2D map (5.16).

Let us finally turn our attention to ordered orbits of N dof Hamiltonian systems(or 2N D symplectic maps), lying on s-dimensional tori with 2 � s � N forHamiltonian flows, and 1 � s � N for maps. For such orbits, all deviation vectorstend to fall on the s-dimensional tangent space of the torus on which the motionlies. Thus, if we start with k � s general deviation vectors, these will remainlinearly independent on the s-dimensional tangent space of the torus, since thereis no particular reason for them to become linearly dependent. As a consequenceGALIk remains practically constant and different from zero for k � s. On the otherhand, GALIk tends to zero for k > s, since some deviation vectors will eventuallyhave to become linearly dependent. In [93, 316] it was shown that the behavior ofGALIk for such orbits is given by

GALIk.t/ /

8<:

constant if 2 � k � s1

tk�s if s < k � 2N � s1

t2.k�N / if 2N � s < k � 2N

: (5.58)

Note that setting s D N in (5.58) we deduce that GALIk remains constant for2 � k � N and decreases to zero as 1=t2.k�N / for N < k � 2N in accordancewith (5.57).

As a final remark we note that the behavior of the GALIs for stable and unstableperiodic orbits, both for Hamiltonian flows and symplectic maps, is studied in [247](see also Problem 5.1).

5.3.2 Numerical Verification and Applications

5.3.2.1 Low-Dimensional Hamiltonian Systems

In order to apply the GALI method to low-dimensional Hamiltonian systems andverify the theoretically predicted behaviors of Sect. 5.3.1, we shall use two simpleexamples: the two dof Henon-Heiles system (5.18)

H2 D 1

2.p2

x C p2y/ C 1

2.x2 C y2/ C x2y � 1

3y3;

and the three dof Hamiltonian


H3 D3X

iD1

!i

2.q2

i C p2i / C q2

1q2 C q21q3; (5.59)

studied in [32,104]. We keep the parameters of the two systems fixed at the energiesH2 D 0:125 and H3 D 0:09, with !1 D 1, !2 D p

2 and !3 D p3. In order

to illustrate the behavior of GALIk, for different values of k, we consider somerepresentative cases of chaotic and regular orbits of the two systems. We note herethat a MAPLE algorithm for the computation of the GALI for orbits of the Henon-Heiles system is presented in Appendix B.

Let us start with a chaotic orbit of the two dof Hamiltonian (5.18), with initialconditions x D 0, y D �0:25, px D 0:42, py D 0. In Fig. 5.8a we see the timeevolution of K1

t (denoted by L1.t/) of this orbit. The computation is carried outuntil K1

t stops having large fluctuations and approaches a positive value (indicatingthe chaotic nature of the orbit), which could be considered as a good approximationof the maximum LCE, L1. Actually, for t � 105, we find L1 � 0:047.

We recall that two dof Hamiltonian systems have only one positive LCE L1,since the second largest is L2 D 0. It also holds that L3 D �L2 and L4 D �L1

and thus formula (5.39), which describes the time evolution of GALIk for chaoticorbits, gives

GALI2.t/ / e�L1t ; GALI3.t/ / e�2L1t ; GALI4.t/ / e�4L1t : (5.60)

a b

Fig. 5.8 (a) The evolution of the two largest finite time LCEs K1t (solid curve), K2

t (dashedcurve) (denoted by L1.t/ and L2.t/ respectively), and L1.t/ � L2.t/ (dotted curve) for a chaoticorbit with initial conditions x D 0, y D �0:25, px ' 0:42081, py D 0 of system (5.18).(b) The evolution of GALI2, GALI3 and GALI4 of the same orbit. The plotted lines correspondto functions proportional to e�L1t (solid line), e�2L1t (dashed line) and e�4L1t (dotted line) forL1 D 0:047. Note that the t–axis is linear. The evolution of the norm of the deviation vectorw.t / (with kw.0/k D 1) used for the computation of L1.t/, is also plotted in (b) (gray curve)(after [315])


In Fig. 5.8b we plot GALIk , k D 2; 3; 4 for the same chaotic orbit as a function oftime t . We plot t in linear scale so that, if (5.60) is valid, the slope of GALI2, GALI3

and GALI4 should approximately be �L1= ln 10, �2L1= ln 10 and �4L1= ln 10

respectively. From Fig. 5.8b we see that lines having precisely these slopes, forL1 D 0:047, approximate quite accurately the computed values of the GALIs.

Let us now study the behavior of GALIk for a regular orbit of the 2D Hamiltonian(5.18) with initial conditions x D 0, y D 0, px D 0:5, py D 0. From (5.57) itfollows that in the case of an ordered orbit of a two dof Hamiltonian system theGALIs should evolve as

GALI2.t/ / constant; GALI3.t/ / 1

t2; GALI4.t/ / 1

t4: (5.61)

From Fig. 5.9, where we plot the time evolution of SALI, GALI2, GALI3 and GALI4

for this ordered orbit, we see that the indices do evolve according to predictions(5.61).

As we have seen in Sect. 5.2 the different behavior of SALI (or GALI2) forordered and chaotic orbits can be successfully used for discriminating betweenregions of order and chaos in various dynamical systems. Figures 5.8 and 5.9 clearlyillustrate that GALI3 and GALI4 tend to zero both for ordered and chaotic orbits,but with very different time rates. We may use this difference to distinguish betweenchaotic and ordered motion following a different approach than SALI or GALI2. Letus illustrate this by considering the computation of GALI4: From (5.60) and (5.61),we expect GALI4 / e�4L1t for chaotic orbits and GALI4 / 1=t4 for ordered ones.These time rates imply that, in general, the time needed for the index to becomezero is much larger for ordered orbits. Thus, instead of simply registering the valueof the index at the end of a given time interval—as we did with SALI (GALI2)in Sect. 5.2—let us record the time, tth, needed for GALI4 to reach a very smallthreshold, e.g. 10�12, and color each initial according to the value of tth.

Fig. 5.9 Time evolution, inlog-log scale, of SALI (graycurves), GALI2, GALI3 andGALI4 for the regular orbit ofHamiltonian (5.18) withinitial conditions x D 0,y D 0, px D 0:5, py D 0.Note that the curves of SALIand GALI2 are very close toeach other and thus cannot bedistinguished. Dashed linescorresponding to particularpower laws are also plotted(after [315])


0.0

0.1

0.2py

0.3

0.4

0.5

-0.4 -0.2 0.0 0.2y

0.4 0.6

100 200 300 400 5000tth

Fig. 5.10 Regions ofdifferent values of the time tthneeded for GALI4 to becomeless than 10�12 on the PSSdefined by x D 0 of the twodof Hamiltonian (5.18)(after [315])

The outcome of this procedure for system (5.18) is presented in Fig. 5.10, whichis similar to Fig. 5.5. Each orbit is integrated up to t D 500 units and if the valueof GALI4, at the end of the integration is larger than 10�12 the corresponding initialcondition is colored by the light gray color used for tth � 400. Thus we can clearlydistinguish in this figure among various “degrees” of chaotic behavior in regionscolored black or dark gray—corresponding to small values of tth—and regions ofordered motion colored light gray, corresponding to large values of tth. At the borderbetween them we find points having intermediate values of tth which belong to the“sticky” chaotic regions.

Let us now study the behavior of the GALIs in the case of the three dofHamiltonian (5.59). Following [32, 104] the initial conditions of the orbits of thissystem are defined by assigning arbitrary values to the positions q1, q2, q3, as wellas the so-called “harmonic energies” E1, E2, E3 related to the momenta throughpi D p

2Ei=!i , i D 1; 2; 3. Chaotic orbits of three dof Hamiltonian systemsgenerally have two positive LCEs, L1 and L2, while L3 D 0. So, for approximatingthe behavior of GALIs according to (5.39), both L1 and L2 are needed. In particular,(5.39) gives

GALI2.t/ / e�.L1�L2/t ; GALI3.t/ / e�.2L1�L2/t ;

GALI4.t/ / e�.3L1�L2/t ; GALI5.t/ / e�4L1t ; GALI6.t/ / e�6L1t :(5.62)

Consider first the chaotic orbit with initial conditions q1 D q2 D q3 D 0,E1 D E2 D E3 D 0:03 of system (5.59). Computing L1, L2 for this orbit as thelong time limits of the finite time LCEs, K1

t , K2t , we present the results in Fig. 5.11a.

The computation is carried out until K1t and K2

t stop having large fluctuations


a b

Fig. 5.11 (a) The evolution of the two largest finite time LCEs, K1t , K2

t (denoted by L1.t/, L2.t/),for the chaotic orbit with initial condition q1 D q2 D q3 D 0, E1 D E2 D E3 D 0:03 of the 3Dsystem (5.59). (b) The evolution of GALIk with k D 2; : : : ; 6, of the same orbit. The plotted linescorrespond to functions proportional to e�.L1�L2/t , e�.2L1�L2/t , e�.3L1�L2/t , e�4L1t and e�6L1t

for L1 D 0:03, L2 D 0:008. Note that the t–axis is linear (after [315])

and approach some positive values (since the orbit is chaotic), which could beconsidered as good approximations of their limits L1, L2. Actually for t � 105

we have L1 � 0:03 and L2 � 0:008. Using these values as good approximationsof L1, L2 we see in Fig. 5.11b that the slopes of all GALIs are well reproducedby (5.62).

Next, we consider the case of ordered orbits of the three dof Hamiltonian system,for which the GALIs should behave as:

GALI2.t/ / constant; GALI3.t/ / constant; GALI4.t/ / 1t2 ;

GALI5.t/ / 1t4 ; GALI6.t/ / 1

t6 :(5.63)

according to (5.57). In order to verify (5.63) we shall follow a specific ordered orbitof system (5.59) with initial conditions q1 D q2 D q3 D 0, E1 D 0:005, E2 D0:085, E3 D 0. The ordered nature of this orbit is revealed by the slow convergenceof its K1

t to zero, implying that L1 D 0 (Fig. 5.12a). In Fig. 5.12b, we plot the valuesof all GALIs of this orbit with respect to time t . From these results we see again thatthe different behaviors of GALIs are very well approximated by (5.57).

From the results of Figs. 5.11 and 5.12, we conclude that in the case of threedof Hamiltonian systems not only GALI2 (or SALI), but also GALI3 has differentbehavior for ordered and chaotic orbits. Hence, the natural question arises whetherGALI3 can be used instead of SALI for the faster detection of chaotic and orderedmotion in three dof Hamiltonians and, by extension, whether GALIk , with k > 3,should be preferred for systems with N > 3. The obvious computational drawback,of course, is that the evaluation of GALIk requires that we numerically follow theevolution of more than two deviation vectors.


a b

Fig. 5.12 (a) The evolution of K1t (denoted by L1.t/) for the ordered orbit with initial condition

q1 D q2 D q3 D 0, E1 D 0:005, E2 D 0:085, E3 D 0 of the system (5.59). (b) The evolution ofGALIk with k D 2; : : : ; 6, of the same orbit. The plotted lines correspond to functions proportionalto 1

t2 , 1t4 and 1

t6 (after [315])

For example the computation of GALI3 for a given time interval t needs moreCPU time than SALI, since we follow the evolution of three deviation vectorsinstead of two. This is particularly true for ordered orbits as the index does notbecome zero and its evolution has to be followed for the whole prescribed timeinterval. In the case of chaotic orbits, however, the situation is different since theindex tends exponentially to zero even faster than GALI2 or SALI. Let us consider,for example, the chaotic orbit of Fig. 5.11. The usual technique to characterize anorbit as chaotic is to check, after some time interval, if its SALI has become lessthan a very small threshold value, e.g. 10�8. For this particular orbit, this thresholdvalue was reached at t � 760. Adopting the same threshold to characterize an orbitas chaotic, we find that GALI3 becomes less than 10�8 after t � 335, requiring onlyas much as 65% of the CPU time needed for SALI to reach the same threshold!

So, using GALI3 instead of SALI, we gain considerably in CPU time for chaoticorbits, while we lose for ordered orbits. Thus, the efficiency of using GALI3 fordiscriminating between chaos and order in a three dof system depends on thepercentage of phase space occupied by chaotic orbits (if all orbits are ordered GALI3

requires more CPU time than SALI). More crucially, however, it depends on thechoice of the final time, up to which each orbit is integrated. As an example, letus integrate, up to t D 1; 000 time units, all orbits whose initial conditions lie ona dense grid in the subspace q3 D p3 D 0, p2 � 0 of a four-dimensional surfaceof section (with q1 D 0) of system (5.59), attributing to each grid point a coloraccording to the value of GALI3 at the end of the integration. If GALI3 of an orbitbecomes less than 10�8 for t < 1; 000 the evolution of the orbit is stopped, itsGALI3 value is registered and the orbit is characterized as chaotic. The outcome ofthis experiment is presented in Fig. 5.13.


0.0-0.4

log(GALI3)-8 -6 -4 -2 0

-0.3 -0.2 -0.1q2

0.10.0 0.2 0.3 0.4

0.1

0.2

p 2

0.3

0.4Fig. 5.13 Regions ofdifferent values of theGALI3 on the subspaceq3 D p3 D 0, p2 � 0 of thefour-dimensional q1 D 0

surface of section of the threedof system (5.59) att D 1; 000 (after [315])

We find that 77% of the orbits of Fig. 5.13 are characterized as chaotic, havingGALI3 < 10�8. In order to have the same percentage of orbits identified aschaotic using SALI (i.e. having SALI < 10�8) the same experiment has to becarried out for t D 2; 000 units, requiring 53% more CPU time. Due to the highpercentage of chaotic orbits, in this case, even when the SALI is computed fort D 1; 000 the corresponding CPU time is 12% higher than the one needed for thecomputation of Fig. 5.13, while only 55% of the orbits are identified as chaotic. Thusit becomes evident that a carefully designed application of GALI3—or GALIk forthat matter—can significantly diminish the computational time needed for a reliablediscrimination between regions of order and chaos in Hamiltonian systems withN > 2 dof.

5.3.2.2 High-Dimensional Hamiltonian Systems

Let us now apply the GALI method to study chaotic, ordered and diffusive motionin multi-dimensional Hamiltonian systems. In particular, we consider the FPU�ˇ

model of N particles with Hamiltonian (3.15)

H DNX

iD1

p2i

2C

NXiD0

�.qiC1 � qi /

2

2C ˇ.qiC1 � qi /

4

4

�; (5.64)

with q1; : : : ; qN being the displacements of the particles with respect to theirequilibrium positions, and p1; : : : ; pN the corresponding momenta. Defining normal


mode variables by (see (4.3))

Qk Dr

2

N C 1

NXiD1

qi sin

�ki�

N C 1

�; Pk D

r2

N C 1

NXiD1

pi sin

�ki�

N C 1

�;

k D 1; : : : ; N; (5.65)

the unperturbed Hamiltonian (5.64) with ˇ D 0 is written as the sum of the so-calledharmonic energies Ei having the form

Ei D 1

2

�P 2

i C !2i Q2

i

; !i D 2 sin

�i�

2.N C 1/

�i D 1; : : : ; N; (5.66)

!i being the corresponding harmonic frequencies, as we have seen in Sect. 4.1. Inour study, we fix the number of particles to N D 8 and the system’s parameterto ˇ D 1:5 and impose fixed boundary conditions q0.t/ D qN C1.t/ D p0.t/ DpN C1.t/ D 0, 8t .

We consider first a chaotic orbit of systems (5.64), having seven positive LCEs,which we compute as the limits for t ! 1 of some appropriate quantities Ki

t , i D1; : : : ; 7, to be L1 � 0:170, L2 � 0:141, L3 � 0:114, L4 � 0:089, L5 � 0:064,L6 � 0:042, L7 � 0:020 (Fig. 5.14a). Of course, in the case of the N D 8 particleFPU�ˇ model (5.64) we have Li D �L17�i for i D 1; : : : ; 8 with L8 D L9 D 0.Using the above computed values as good approximations of the real LCEs, we seein Fig. 5.14b, c that the slopes of all GALIk indices are well reproduced by (5.39).

a b c

Fig. 5.14 (a) The time evolution of quantities Kit (denoted by Li ), i D 1; : : : ; 7, having as limits

for t ! 1 the seven positive LCEs Li , i D 1; : : : ; 7, for a chaotic orbit with initial conditionsQ1 D Q4 D 2, Q2 D Q5 D 1, Q3 D Q6 D 0:5, Q7 D Q8 D 0:1, Pi D 0, i D 1; : : : ; 8 of theN D 8 particle FPU�ˇ lattice (5.64). The time evolution of the corresponding GALIk is plottedin (b) for k D 2; : : : ; 6 and in (c) for k D 7; 8; 10; 12; 14; 16. The plotted lines in (b) and (c)correspond to exponentials that follow the asymptotic laws (5.39) for L1 D 0:170, L2 D 0:141,L3 D 0:114, L4 D 0:089, L5 D 0:064, L6 D 0:042, L7 D 0:020. Note that t–axis is linear andthat the slope of each line is written explicitly in (b) and (c) (after [316])


a b c

Fig. 5.15 (a) The time evolution of harmonic energies Ei , i D 1; : : : ; 8, for an ordered orbit withinitial conditions q1 D q2 D q3 D q8 D 0:05, q4 D q5 D q6 D q7 D 0:1, pi D 0, i D 1; : : : ; 8,of the N D 8 particle FPU�ˇ lattice (5.64). The time evolution of the corresponding GALIk isplotted in (b) for k D 2; : : : ; 8, and in (c) for k D 10; 12; 14; 16. The plotted lines in (c) correspondto functions proportional to t�4, t�8, t�12 and t�16, as predicted in (5.57) (after [316])

Turning now to the case of an ordered orbit of system (5.64), we plot inFig. 5.15a the evolution of its harmonic energies Ei , i D 1; : : : ; 8. The harmonicenergies remain practically constant, exhibiting some feeble oscillations, implyingthe regular nature of the orbit. In Fig. 5.15b and c we plot the GALIs of this orbitand verify that their behavior is well approximated by the asymptotic formula (5.57)for N D 8. Note that the GALIk for k D 2; : : : ; 8 (Fig. 5.15b) remain different fromzero, implying that the orbit is indeed quasiperiodic and lies on an eight-dimensionaltorus. In particular, after some initial transient time, they start oscillating aroundnon-zero values whose magnitude decreases with increasing k. On the other hand,the GALIk with 8 < k � 16 (Fig. 5.15c) tend to zero following power law decaysin accordance with (5.57).

From the results of Figs. 5.14 and 5.15, we conclude that the different behaviorof GALIk for chaotic (exponential decay) and ordered orbits (non-zero values orpower law decay) allows for a fast and clear discrimination between the two cases.Consider for example GALI8 which tends exponentially to zero for chaotic orbits(Fig. 5.14c), while it remains small but different from zero in the case of regularorbits (Fig. 5.15b). At t � 150 GALI8 has values that differ almost 35 orders ofmagnitude being GALI8 � 10�36 for the chaotic orbit, while GALI8 � 10�1 forthe regular one. This huge difference in the values of GALI8 clearly discriminatesbetween the two cases.

What happens, however, if we choose an orbit that starts near a torus butslowly drifts away from it, presumably through a thin chaotic layer of higher orderresonances? This phenomenon is recognized by the GALIs, which provide earlypredictions that may be quite relevant for applications. To see this let us choose againinitial conditions for our 8-particle FPU�ˇ model, such that the motion appearsquasiperiodic, with its energy recurring between the modes E1, E3, E5 and E7

(Fig. 5.16a). In particular, we consider an orbit with initial conditions Q1 D 2,


a b c

Fig. 5.16 The time evolution of harmonic energies (a) E1, E3, E5, E7 and (b) E6, E8, for aslowly diffusing orbit of the N D 8 particle FPU�ˇ lattice (5.64). (c) The time evolution of thecorresponding GALIk for k D 2; : : : ; 8, clearly exhibits exponential decay already at t � 10; 000

(after [316])

Q7 D 0:44, Q3 D Q4 D Q5 D Q6 D Q8 D 0, Pi D 0, i D 1; : : : ; 8, having totalenergy H D 0:71.

This orbit, however, is not quasiperiodic, as it drifts away from the initialfour-dimensional torus, exciting new frequencies and sharing its energy with moremodes, after about t D 20; 000 time units. This becomes evident in Fig. 5.16b wherewe plot the evolution of E6 and E8. We see that these harmonic energies, which wereinitially zero, start having non-zero values at t � 20; 000 and exhibit from then onsmall oscillations (note the different scales of ordinate axis of Fig. 5.16a, b), whichlook very regular until t � 66; 000. At that time the values of all harmonic energieschange dramatically, clearly indicating the chaotic nature of the orbit. As we see inFig. 5.16c, this type of diffusion is predicted by the exponential decay of all GALIk,shown already at about t D 10;000.

5.3.2.3 Symplectic Maps

We can also demonstrate the validity of the theoretical predictions of Sect. 5.3.1 inthe case of symplectic maps, by considering for example the 6D map

x01 D x1 C x0

2

x02 D x2 C K

2�sin.2�x1/ � ˇ

2�fsinŒ2�.x5 � x1/� C sinŒ2�.x3 � x1/�g

x03 D x3 C x0

4

x04 D x4 C K

2�sin.2�x3/ � ˇ


x05 D x5 C x0

6

x06 D x6 C K

2�sin.2�x5/ � ˇ


(5.67)

which consists of three coupled standard maps [180] and is a typical nonlinearsystem, in which regions of chaotic and quasi-periodic dynamics are found to


100500-16

-14 GALI5

GALI4

GALI2 GALI5

GALI4

GALI3

GALI2

GALI6GALI3GALI6

-12

-10

-8

Log(

GA

LIs)

-6

-4

-8

-4

-2

-0

-2

slope= - (σ1-σ2)/ In(10)

slope= - (2σ1-σ2-σ3)/ In(10)

slope= - (3σ1-σ2)/ In(10)

slope= - (4σ1/ In(10)

slope= - (6σ1/ In(10)

150

n

200 250 300-16

-12

slope=-4slope=-2

slope=-6

-0

Log(

GA

LIs)

1 2 3 4

Log(n)

5 6

a b

Fig. 5.17 The evolution of GALIk , k D 2; : : : ; 6, with respect to the number of iteration n for(a) the chaotic orbit C1 and (b) the ordered orbit R1 of the 6D map (5.67). The plotted linescorrespond to functions proportional to e�.L1�L2/n, e�.2L1�L2�L3/n , e�.3L1�L2/n , e�4L1n, e�6L1n

for L1 D 0:70, L2 D 0:57, L3 D 0:32 in (a) and proportional to n�2, n�4, n�6 in (b) (after [244])

coexist. Note that each coordinate is given modulo 1 and that in our study we fix theparameters of the map (5.67) to K D 3 and ˇ D 0:1.

In order to verify numerically the validity of (5.39) and (5.57), we study twotypical orbits of map (5.67), a chaotic one with initial condition x1 D x3 D x5 D0:8, x2 D 0:05, x4 D 0:21, x6 D 0:01 (orbit C1) and an ordered one with initialcondition x1 D x3 D x5 D 0:55, x2 D 0:05, x4 D 0:01, x6 D 0 (orbit R1) lying ona three-dimensional torus. In Fig. 5.17 we see the evolution of GALIk, k D 2; : : : ; 6,for these two orbits.

For a chaotic orbit of the 6D map (5.67) we have L1 D �L6, L2 D �L5,L3 D �L4 with L1 � L2 � L3 � 0, since the LCEs are ordered in pairs ofopposite signs. So for the evolution of GALIk (5.39) gives

GALI2.n/ / e�.L1�L2/n; GALI3.n/ / e�.2L1�L2�L3/n;

GALI4.n/ / e�.3L1�L2/n; GALI5.n/ / e�4L1n; GALI6.n/ / e�6L1n:(5.68)

The positive LCEs of the chaotic orbit C1 were found to be L1 � 0:70, L2 � 0:57,L3 � 0:32. From the results of Fig. 5.17a we see that the exponentials in (5.68)for L1 D 0:70, L2 D 0:57, L3 D 0:32 approximate quite accurately the computedvalues of GALIs.

According to (5.57), the behavior of GALIk, k D 2; : : : ; 6 for the regular orbitR1 is given by

GALI2.n/ / constant; GALI3.n/ / constant; GALI4.n/ / 1n2 ;

GALI5.n/ / 1n4 ; GALI6.n/ / 1

n6 :(5.69)


From the results of Fig. 5.17b we see that these approximations also describe verywell the evolution of GALIs.

5.3.2.4 Motion on Low-Dimensional Tori

Let us now turn our attention to an important feature of the GALI method: itsability to detect ordered motion on low-dimensional tori. In order to illustrate thisfeature of the index we consider again the FPU�ˇ lattice (5.64) as a toy model,although the capability of the GALI method to identify motion on low-dimensionaltori was also observed for multi-dimensional symplectic maps [63]. Selecting initialconditions such that only a small number of the harmonic energies Ei (5.66) ofthe FPU�ˇ model are initially excited, we observe at small enough energies thatthe system exhibits the famous FPU recurrences, whereby energy is exchangedquasiperiodically only between the excited Ei (see Chap. 6).

In particular, choosing an orbit with initial conditions qi D 0:1, pi D 0, i D 1;

: : : ; 8, and total energy H D 0:01 we distribute the energy among 4 modes, Ei ,i D 1; 3; 5; 7 in the 8-particle FPU�ˇ lattice with ˇ D 1:5. In Fig. 5.18a we observethat actually only 4 modes are excited, while in Fig. 5.18b we see that only theGALIk for k D 2; 3; 4 remain constant, implying that the motion lies on afour-dimensional torus. All the higher order GALIs in Fig. 5.18b, c decay by powerlaws, whose exponents are obtained from (5.58) for N D 8 and s D 4.

Thus, for an ordered orbit (5.58) implies that the largest order k of its GALIsthat eventually remains constant determines the dimension of the torus on whichthe motion occurs. Instead of taking particular initial conditions which lie on low-dimensional tori, let us perform a more global investigation of the FPU�ˇ model,aiming to track more “globally” the location of such tori. For this purpose, we

a b c

Fig. 5.18 (a) The time evolution of harmonic energies Ei for an ordered orbit lying on afour-dimensional torus of the N D 8 particle FPU�ˇ lattice (5.64). Recurrences occur betweenE1, E3, E5 and E7, while all other harmonic energies remain practically zero. The time evolution ofthe corresponding GALIk is plotted in (b) for k D 2; : : : ; 8 and in (c) for k D 9; 11; 13; 14; 16. Theplotted lines in (b) and (c) correspond to the precise power laws predicted by (5.58) (after [316])


consider the Hamiltonian system (5.64) with N D 4, for which ordered motioncan occur on an s-dimensional torus with s D 2, 3, 4. According to (5.58) thecorresponding GALIs of order k � s will be constant, while the remaining oneswill tend to zero following particular power laws. Thus, following [147], in order tolocate low-dimensional tori we compute the GALIk, k D 2; 3; 4 in the subspace.q3; q4/ of the system’s phase space, considering orbits with initial conditionsq1 D q2 D 0:1, p1 D p2 D p3 D 0, while p4 is computed to keep the totalenergy H constant at H D 0:010075.

Since the constant final values of GALIk, k D 2; : : : ; s, decrease with increasingorder k (see for example the GALIs with 2 � k � 8 in Fig. 5.15), we chose to“normalize” the values of GALIk , k D 2; 3; 4 of each individual initial condition, bydividing them by the largest GALIk value, max.GALIk/, obtained from all studiedorbits. Thus, in Fig. 5.19 we color each initial condition according to its “normalizedGALIk” value

gk D GALIk

max.GALIk/: (5.70)

In each panel of Fig. 5.19, large gk values (colored in yellow or in light red)correspond to initial conditions whose GALIk eventually stabilizes to constant, non-zero values. On the other hand, darker regions correspond to small gk values, whichcorrespond to power law decays of GALIs.

Consequently, motion on two-dimensional tori, which corresponds to large finalGALI2 values and small final GALI3 and GALI4 values, should be located inareas of the phase space colored yellow or light red in Fig. 5.19a, and in black inFig. 5.19b, c. A region of the phase space with these characteristics is for examplelocated on the upper border of the colored areas of Fig. 5.19. A particular initialcondition with q3 D 0:106, q4 D 0:0996 in this region is denoted by a triangle in all

0.00 0.05 0.10q3 q3q3

0.00

0.05

0.10

q 4

0.00 0.05 0.10

g4g3g2

0.00 0.05 0.10

10-1

10-2

10-3

10-4

100

gk

a b c

Fig. 5.19 Regions of different gk (5.70) values, k D 2, 3, 4, on the .q3; q4/ plane of theHamiltonian system (5.64) with N D 4. Each initial condition is integrated up to t D 106 , andcolored according to its final (a) g2, (b) g3, and (c) g4 value, while white regions correspondto forbidden initial conditions. Three particular initial conditions of ordered orbits on 2-, 3- andfour-dimensional tori are marked by a triangle, a square and a circle respectively (after [147])


-6

-4

-2

0

1 2 3 4 5

log 1

0 G

ALI

s

log10t log10tlog10t

2d torus

GALI2GALI3GALI4

1 2 3 4 5

3d torus

1 2 3 4 5

4d torusa b c

Fig. 5.20 The time evolution of GALIs for ordered orbits lying on a (a) two-dimensional torus,(b) three-dimensional torus, and (c) four-dimensional torus of the Hamiltonian system (5.64) withN D 4. The initial conditions of these orbits are marked respectively by a triangle, a square and acircle in Fig. 5.19 (after [147])

panels of Fig. 5.19. This orbit indeed lies on a two-dimensional torus, as we see fromthe evolution of its GALIs shown in Fig. 5.20a. In a similar way, an ordered orbit ona three-dimensional torus should be located in regions colored in black only in theg4 plot of Fig. 5.19c. An orbit of this type is the one with q3 D 0:085109, q4 D 0:054

denoted by a square in Fig. 5.19, which actually evolves on a three-dimensionaltorus, as only its GALI2 and GALI3 remain constant (Fig. 5.20b). Finally, the orbitwith q3 D 0:025, q4 D 0 (denoted by a circle in Fig. 5.19) inside a region of thephase space colored in yellow or light red in all panels of Fig. 5.19, is an orderedorbit on a four-dimensional torus and its GALI2, GALI3 and GALI4 remain constant(Fig. 5.20c). It is worth noting, that chaotic motion would lead to very small GALIk

and gk values, since all GALIs would tend to zero exponentially, and consequentlywould correspond to regions colored in black in all panels of Fig. 5.19. Thus, chaoticmotion can be easily distinguished from regular motion on low-dimensional tori.

From the results of Figs. 5.19 and 5.20 it becomes evident that the comparisonof color plots of “normalized GALIk” values can facilitate the search for low-dimensional tori in multi-dimensional phase spaces.

Appendix A: Wedge Product

We present here an introduction to the theory of wedge products following [310,315]and textbooks such as [159, 321]. Let us consider an M -dimensional vector spaceV over the field of real numbers R. The exterior algebra of V is denoted by .V /

and its multiplication, known as the wedge product (or exterior product), is writtenas ^. The wedge product is associative:

.u ^ v/ ^ w D u ^ .v ^ w/: (5.71)


for u; v; w 2 V and bilinear

.c1u C c2v/ ^ w D c1.u ^ w/ C c2.v ^ w/;

w ^ .c1u C c2v/ D c1.w ^ u/ C c2.w ^ v/; (5.72)

for u; v; w 2 V and c1; c2 2 R. The wedge product on V also satisfies

u ^ u D 0; (5.73)

for all vectors u 2 V . Thus we have that

u ^ v D �v ^ u; (5.74)

for all vectors u; v 2 V and

u1 ^ u2 ^ � � � ^ uk D 0; (5.75)

whenever u1; u2; : : : ; uk 2 V are linearly dependent. Elements of the form u1 ^u2 ^� � � ^ uk with u1; u2; : : : ; uk 2 V are called k-vectors.

Let f Oe1; Oe2; : : : ; OeM g be an orthonormal basis of V , i.e. Oei , i D 1; 2; : : : ; M , arelinearly independent vectors of unit magnitude and

Oei � Oej D ıij (5.76)

where (�) denotes the inner product in V and

ıij D

1 for i D j

0 for i ¤ j: (5.77)

It can be easily seen that the set

f Oei1 ^ Oei2 ^ � � � ^ Oeik j 1 � i1 < i2 < � � � < ik � M g (5.78)

is a basis of k.V / since any wedge product of the form u1 ^ u2 ^ � � � ^ uk can bewritten as a linear combination of the k-vectors of (5.78).

The coefficients of a k-vector u1 ^ u2 ^ � � � ^ uk are the minors of the matrix C

having as rows the coordinates of vectors ui , i D 1; 2; : : : ; k, with respect to theorthonormal basis Oei , i D 1; 2; : : : ; M . In matrix form we have

26664

u1

u2

:::

uk

37775 D

26664

u11 u12 � � � u1M

u21 u22 � � � u2M

::::::

:::

uk1 uk2 � � � ukM

37775 �

26664

Oe1

Oe2

:::

OeM

37775 D C �

26664

Oe1

Oe2

:::

OeM

37775 ; (5.79)


where uij , i D 1; 2; : : : ; k, j D 1; 2; : : : ; M are real numbers. Then the wedgeproduct u1 ^ u2 ^ � � � ^ uk is written as

u1 û2 ^� � �ûk DX

1�i1<i2<��<ik�M

ˇˇˇˇˇ

u1i1 u1i2 � � � u1ik

u2i1 u2i2 � � � u2ik:::

::::::

uki1 uki2 � � � ukik

ˇˇˇˇˇ

Oei1 ^ Oei2 ^� � �^ Oeik ; (5.80)

where the sum is performed over all possible combinations of k indices out of theM total indices, and j � j denotes the determinant. The norm of the k-vector u1 û2 ^ � � � ^ uk is given by

ku1 ^ u2 ^ � � � ^ ukk D

8ˆ<ˆ:

X1�i1<i2<��<ik�M

ˇˇˇˇˇ

u1i1 u1i2 � � � u1ik

u2i1 u2i2 � � � u2ik:::

::::::

uki1 uki2 � � � ukik

ˇˇˇˇˇ

29>>>>=>>>>;

1=2

: (5.81)

Appendix B: Example Algorithms for the Computationof the SALI and GALI Chaos Indicators

We present here MAPLE algorithms for the computation of the SALI and GALIs.As a test model we consider the Henon-Heiles system (5.18) and compute theindices for the chaotic orbit of Figs. 5.3 and 5.8. We have included several commentsin the algorithms in order to facilitate their modification for other systems.

A MAPLE Algorithm for the Computation of the SALI

The following algorithm uses (5.13) for the computation of the SALI.

> restart:> with(LinearAlgebra):> Digits:=20:

Initialization:The Henon-Heiles Hamiltonian (5.18)

> H := 1/2*(pxˆ2 + pyˆ2) + 1/2*(xˆ2 + yˆ2) + xˆ2*y - 1/3*yˆ3;

Parameters:Degrees of freedom

> dof := 2;


Variables for the state of the system and the variations> q := Vector(dof, [x, y]);> p := Vector(dof, [px, py]);> deltaq := Matrix(dof, 2, [[dx1, dx2], [dy1, dy2]]);> deltap := Matrix(dof, 2, [[dpx1, dpx2], [dpy1, dpy2]]);

Time range of the integration and time increment> T0 := 0.0;> Tend := 1.0e3;> Tinc := 1.0;

Initial conditions and initial orthonormal deviation vectors> ics := x(0)=0.0, y(0)=-0.25, px(0)=0.42081, py(0)=0.0,> dx1(0)=1.0, dy1(0)=0.0, dpx1(0)=0.0, dpy1(0)=0.0,> dx2(0)=0.0, dy2(0)=1.0, dpx2(0)=0.0, dpy2(0)=0.0;

Matrix used for the computation of the SALI> N := floor((Tend-T0)/Tinc);> SALI := Matrix(N, 2, datatype=float[8], []);

Equations of motion> for i from 1 to 2 do> deq[i] := diff(q[i](t), t) = diff(H, p[i]);> deq[2+i] := diff(p[i](t), t) = -diff(H, q[i]):> end do:

Variational equations> for nc from 1 to 2 do> k := 5 + 4*(nc-1):> for nr from 1 to 2 do> deq[k] := diff(deltap[nr,nc](t),t) => -add(diff(H,q[nr],q[j])*deltaq[j,nc],j=1..dof);> deq[k+2] := diff(deltaq[nr,nc](t),t) => add(diff(H,p[nr],p[j])*deltap[j,nc],j=1..dof);> k:=k+1;> end do:> end do:

Make all equations explicit functions of time> eqs := seq(lhs(deq[i]) = subs(> f seq(q[i]=q[i](t),i=1..dof),> seq(p[i]=p[i](t),i=1..dof),> seq(seq(deltaq[i,j]=deltaq[i,j](t),i=1..dof),j=1..2),> seq(seq(deltap[i,j]=deltap[i,j](t),i=1..dof),j=1..2)g,> rhs(deq[i])), i = 1..12);

The integration> for i from 1 to N do> T := evalf(T0+i*Tinc):> # integrate the system of ODEs numerically> dsn := dsolve(feqs,icsg,numeric,range = T-Tinc..T);> # the 2 variational vectors> v1 := Vector(4,[> subs(dsn(T),dx1(t)),


> subs(dsn(T),dy1(t)),> subs(dsn(T),dpx1(t)),> subs(dsn(T),dpy1(t))]);> v2 := Vector(4,[> subs(dsn(T),dx2(t)),> subs(dsn(T),dy2(t)),> subs(dsn(T),dpx2(t)),> subs(dsn(T),dpy2(t))]);> # renormalize these vectors> v1 := v1/sqrt(v1.v1):> v2 := v2/sqrt(v2.v2):> tplus := sqrt((v1+v2).(v1+v2)):> tminus := sqrt((v1-v2).(v1-v2)):> # compute the SALI using Eq. (5.13)> SALI[i,1] := T:> SALI[i,2] := min(tplus,tminus);> # use the result as new initial conditions for the next step> ics:= x(T)=subs(dsn(T),x(t)), y(T)=subs(dsn(T),y(t)),> px(T)=subs(dsn(T),px(t)), py(T)=subs(dsn(T),py(t)),> dx1(T)=v1[1], dy1(T)=v1[2], dpx1(T)=v1[3], dpy1(T)=v1[4],> dx2(T)=v2[1], dy2(T)=v2[2], dpx2(T)=v2[3], dpy2(T)=v2[4];> end do:

A MAPLE Algorithm for the Computation of the GALI

The following algorithm computes the whole spectrum of the GALIs using theSingular Value Decomposition procedure of an appropriate matrix described inSect. 5.3. In particular it implements (5.29) for the evaluation of GALIs.

> restart:> with(LinearAlgebra):> Digits := 20:

Initialization:The Henon-Heiles Hamiltonian (5.18)

> H := 1/2*(pxˆ2 + pyˆ2) + 1/2*(xˆ2 + yˆ2) + xˆ2*y - 1/3*yˆ3;

Parameters:Degrees of freedom

> dof := 2;

Variables for the state of the system and the variations> q := Vector(dof, [x, y]);> p := Vector(dof, [px, py]);> deltaq := Matrix(dof, 4, [[dx1,dx2,dx3,dx4],> [dy1,dy2 dy3,dy4]]);> deltap := Matrix(dof, 4, [[dpx1,dpx2,dpx3,dpx4],> [dpy1,dpy2,dpy3,dpy4]]);

Time range of the integration and time increment> T0 := 0.0;


> Tend := 1.0e3;> Tinc := 1.0;

Initial conditions and initial orthonormal deviation vectors> ics := x(0)=0, y(0)=-0.25, px(0)=0.42081, py(0)=0.0,> dx1(0)=1.0, dy1(0)=0.0, dpx1(0)=0.0, dpy1(0)=0.0,> dx2(0)=0.0, dy2(0)=1.0, dpx2(0)=0.0, dpy2(0)=0.0,> dx3(0)=0.0, dy3(0)=0.0, dpx3(0)=1.0, dpy3(0)=0.0,> dx4(0)=0.0, dy4(0)=0.0, dpx4(0)=0.0, dpy4(0)=1.0;

Matrix used for the computation of the GALI> N := floor((Tend-T0)/Tinc);> GALI := Matrix(N, 4, datatype=float[8], []);

Equations of motion> for i from 1 to 2 do> deq[i] := diff(q[i](t), t) = diff(H, p[i]);> deq[2+i] := diff(p[i](t), t) = -diff(H, q[i]):> end do:

Variational equations> for nc from 1 to 4 do> k := 5 + 4*(nc-1):> for nr from 1 to 2 do> deq[k] := diff(deltap[nr,nc](t),t) => -add(diff(H,q[nr],q[j])*deltaq[j,nc],j=1..dof);> deq[k+2] := diff(deltaq[nr,nc](t),t) => add(diff(H,p[nr],p[j])*deltap[j,nc],j=1..dof);> k:=k+1;> end do:> end do:

Make all equations explicit functions of time> eqs := seq(lhs(deq[i]) = subs(> f seq(q[i]=q[i](t),i=1..dof),> seq(p[i]=p[i](t),i=1..dof),> seq(seq(deltaq[i,j]=deltaq[i,j](t),i=1..dof),j=1..4),> seq(seq(deltap[i,j]=deltap[i,j](t),i=1..dof),j=1..4)g,> rhs(deq[i])), i = 1..20);

The integration> for i from 1 to N do> T := evalf(T0+i*Tinc):> # integrate the system of ODEs numerically> dsn := dsolve(feqs,icsg,numeric,range=T-Tinc..T);> # the 4 deviation vectors> v1 := Vector(4,[> subs(dsn(T),dx1(t)), subs(dsn(T),dy1(t)),> subs(dsn(T),dpx1(t)), subs(dsn(T),dpy1(t))]);> v2 := Vector(4,[> subs(dsn(T),dx2(t)), subs(dsn(T),dy2(t)),> subs(dsn(T),dpx2(t)), subs(dsn(T),dpy2(t))]);> v3 := Vector(4,[> subs(dsn(T),dx3(t)), subs(dsn(T),dy3(t)),> subs(dsn(T),dpx3(t)), subs(dsn(T),dpy3(t))]);


> v4 := Vector(4,[> subs(dsn(T),dx4(t)), subs(dsn(T),dy4(t)),> subs(dsn(T),dpx4(t)), subs(dsn(T),dpy4(t))]);> # renormalize these vectors> v1 := v1/sqrt(v1.v1):> v2 := v2/sqrt(v2.v2):> v3 := v3/sqrt(v3.v3):> v4 := v4/sqrt(v4.v4):> # fill the columns of matrix A with these vectors> A := Matrix(4,4,[]):> A[1..4,1] := v1:> A[1..4,2] := v2:> A[1..4,3] := v3:> A[1..4,4] := v4:> # compute GALI_k, k=2,3,4, using Eq. (5.29)> for k from 2 to 4 do> sv:=SingularValues(A[1..4,1..k],output=’list’):> GALI[i,k] := mul(sv[i],i=1..k):> end do:> GALI[i,1] := T:> # use the result as new initial conditions for the next step> ics:= x(T)=subs(dsn(T),x(t)), y(T)=subs(dsn(T),y(t)),> px(T)=subs(dsn(T),px(t)), py(T)=subs(dsn(T),py(t)),> dx1(T)=v1[1], dy1(T)=v1[2], dpx1(T)=v1[3], dpy1(T)=v1[4],> dx2(T)=v2[1], dy2(T)=v2[2], dpx2(T)=v2[3], dpy2(T)=v2[4],> dx3(T)=v3[1], dy3(T)=v3[2], dpx3(T)=v3[3], dpy3(T)=v3[4],> dx4(T)=v4[1], dy4(T)=v4[2], dpx4(T)=v4[3], dpy4(T)=v4[4];> end do:

Exercises

Exercise 5.1. Consider a 2N -dimensional vector space over the field of realnumbers R, which has the usual Euclidean norm and is spanned by the usualorthonormal basis f Oe1; Oe2; : : : ; Oe2N g with Oe1 D .1; 0; 0; : : : ; 0/, Oe2 D .0; 1; 0; : : : ; 0/,: : :, Oe2N D .0; 0; 0; : : : ; 1/. Consider also two unit vectors Ow1, Ow2 in this space so that

Ow1 D2NXiD1

w1i Oei ; Ow2 D2NXiD1

w2i Oei ; (5.82)

andP2N

iD1 w21i D 1,

P2NiD1 w2

2i D 1. Then, for the 2-vector Ow1 ^ Ow2 of (5.80) and itsnorm from (5.81), show the validity of (5.23)

k Ow1 ^ Ow2k D 1

2k Ow1 � Ow2k � k Ow1 C Ow2k:

Hint: Consult Appendix B of [315].


Exercise 5.2. Consider the k�2N matrix W , (5.27), having as rows the coordinatesof k unit deviation vectors Owi , i D 1; 2; : : : ; k. According to (5.24) GALIk is equalto the “volume” vol.Pk/ of the parallelepiped Pk having as edges these vectors. Thisvolume is also given by vol.Pk/ D pjW � W Tj, where j � j denotes the determinantof a matrix and .T/ its transpose (see for instance [174, Chap. 5]). Following forexample [277, Sect. 2.6], perform the Singular Value Decomposition of W T andwrite this matrix as a product of a 2N � k column-orthogonal matrix U , a k � k

diagonal matrix Z with positive or zero elements zi , i D 1; : : : ; k (the so-calledsingular values), and the transpose of a k � k orthogonal matrix V . Then showthat the GALIk is equal to the product of these singular values (5.29). Hint: Consultresults presented in [316].

Exercise 5.3. (a) Show that a deviation vector which is initially located in thetangent space of an N -dimensional torus will remain there forever.

(b) Consider an ordered orbit of an N dof Hamiltonian system lying on aN -dimensional torus, and k linearly independent deviation vectors such thatm (m � N and m � k) of them are initially located in the tangent space of thetorus. Show that for this choice of deviation vectors the evolution of GALIk is

GALIk.t/ /

8<:

constant if 2 � k � N1

t2.k�N /�m if N < k � 2N and 0 � m < k � N1

tk�N if N < k � 2N and m � k � N

:

Hint: Consult results presented in [315].

Exercise 5.4. Following an analysis similar to the one presented in Sect. 5.3.1.2show that the time evolution of GALIk for ordered motion on s-dimensional toriwith 2 � s � N for N dof Hamiltonian flows, and 1 � s � N for 2N D maps isgiven by (5.58)

GALIk.t/ /

8<:

constant if 2 � k � s1

tk�s if s < k � 2N � s1

t2.k�N / if 2N � s < k � 2N

:

Hint: Consult results presented in [93] and [316].

Problems

Problem 5.1. The Behavior of GALI for Periodic Orbits.

(a) Unstable periodic orbits have at least one positive LCE (L1 > 0), which impliesthat nearby orbits diverge exponentially from them. Applying the analysispresented in Sect. 5.3.1.1 for chaotic orbits which also have L1 > 0, show thatGALIk of unstable periodic orbits tends to zero exponentially following the law


GALIk.t/ / e�Œ.L1�L2/C.L1�L3/C��C.L1�Lk/�t : (5.83)

(b) Following an analysis similar to the one presented in Sect. 5.3.1.2 show thatthe time evolution of GALIk for stable periodic orbits of N dof Hamiltoniansystems is given by

GALIk /(

1

tk�1 if 2 � k � 2N � 11

t2N if k D 2N: (5.84)

(c) Stable periodic orbits of symplectic maps correspond to stable fixed pointsof the map, which are located inside islands of stability. Observing that anydeviation vector from the stable periodic orbit performs a rotation around thefixed point show that

GALIk / constant; 2 � k � 2N: (5.85)

for stable periodic orbits of 2N D symplectic maps.Hint: Consult results presented in [247].

Problem 5.2. Connection Between the Dynamics of Hamiltonian Flows and Sym-plectic Maps. N dof Hamiltonian systems can be considered as dynamicallyequivalent to 2.N � 1/D maps, since the latter can be interpreted as appropriatesurfaces of section of the former. Despite this relation the GALIs behave differentlyfor flows and maps. According to (5.85) GALIs remain constant for stable periodicorbits of maps, while they decrease to zero for flows according to (5.84).

In order to understand this difference, consider as an example the Henon-Heilessystem (5.18). Compute for a stable periodic orbit of this system, GALI2 and GALI3,using not the whole deviation vectors but their components orthogonal to the flow.For a two dof Hamiltonian system it is not necessary to compute GALI4 with thesevectors since GALI4 D 0 (why?). Check whether the computed GALIs followpredictions (5.84) or (5.85). Hint: Consult results presented in [247].

Chapter 6FPU Recurrences and the Transitionfrom Weak to Strong Chaos

Abstract The present chapter starts with a historical introduction to the FPU one-dimensional lattice as it was first integrated numerically by Fermi Pasta and Ulamin the 1950s and describes the famous paradox of the FPU recurrences. First wereview some of the more recent attempts to explain it based on the nonlinear normalmodes (NNMs) representing continuations of the lowest q D 1; 2; 3; : : : modesof the linear problem, termed q-breathers by Flach and co-workers, due to theirexponential localization in Fourier space. We then present an extension of thisapproach focusing on certain low-dimensional, so-called q-tori, in the neighborhoodof these NNMs, which are also exponentially localized and can be constructedby Poncare-Linstedt series. We demonstrate how q-tori reconcile q-breathers withthe “natural packets” approach of Berchialla and co-workers. Finally, we use theGALI method to determine the destabilization threshold of q-tori and study the slowdiffusion of chaotic orbits associated with the breakdown of the FPU recurrences.

6.1 The Fermi Pasta Ulam Problem

6.1.1 Historical Remarks

In the history of Hamiltonian dynamics few discoveries have inspired so manyresearchers and motivated so many theoretical and numerical studies as the oneannounced in 1955, by E. Fermi, J. Pasta and S. Ulam (FPU) [71, 121, 133].They had the brilliant idea to use the computers available at that time at the LosAlamos National Laboratory to integrate the ODEs of a chain of 31 identicalharmonic oscillators, coupled to each other by cubic nearest neighbor interactions,to investigate how energy was shared by all particles, as soon as the nonlinear forceswere turned on. To this end, they imposed fixed boundary conditions and solvednumerically the equations of motion


133

134 6 FPU Recurrences and the Transition from Weak to Strong Chaos

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

t

Ek

1

2

3 3

4

4

5 5 5

5

1

2 2

Fig. 6.1 The FPU experiment: Time-integration of the FPU system until just after thefirst approximate recurrence of the initial state. Shown here is how the energy Eq D12

�Pa2

q C 2a2q sin2

�q�

2.N C1/

��is divided over the first five modes aq D PN

kD1 xk sin�

kq�

N C1

�. The

numbers in the figure indicate the wave-number q (after [53]) (The figure is a reproduction of Fig. 1of [121])

Rxk D .xkC1 � 2xk C xk�1/ C ˛h.xkC1 � xk/2 � .xk � xk�1/2

i; k D 1; : : : ; 31;

(6.1)

with x0 D x32 D 0, Px0 D Px32 D 0, where xk D xk.t/ represents each particle’sdisplacement from equilibrium and dots denote differentiation with respect totime t. To their great surprise, starting with the initial condition xk D sin

��k32

�,

corresponding to the first (q D 1) linear normal mode, they observed for smallenergies (˛ D 0:25) a remarkable near-recurrence of the solutions of (6.1) to thisinitial condition after a relatively short time period (see Fig. 6.1 below).

This was amazing! Equilibrium statistical mechanics predicted that all highermodes of oscillation would also be excited and eventually equally share the energyof the system, achieving a state of thermodynamic equilibrium. Instead, FPUobserved that a very small set of modes participated in the dynamics and thesystem practically returned to its initial state after relatively short time periods.It is, therefore, no wonder that this so-called “FPU paradox” created a greatexcitement within the scientific community and generated a number of questionswhich remained open for many years: Could it be, for example, that low energyrecurrences are due to the fact that (6.1) is close to an integrable system? Is itperhaps because this type of nonlinearity is not sufficient for energy equipartition?How can this “energy localization” be explained? A new era of complex phenomenain Hamiltonian dynamics was in the making.

6.1 The Fermi Pasta Ulam Problem 135

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

2.5

3

x

u

Fig. 6.2 The evolution of the initial state u D cos .�x/ in the Korteweg-de Vries equation, at thetimes t D 0 (dotted line) t D 1=� (dashed line). At an intermediate stage (t D 3:6=� , solid line),the solitons are maximally separated, while at t � 30:4=� the initial state is nearly recovered(after [53]) (The figure is a reproduction of Fig. 1 of [352])

In 1965, attempting to explain the results of FPU, Zabusky and Kruskal [352]derived the Korteweg-de Vries (KdV) partial differential equation of shallow waterwaves in the long wavelength and small amplitude approximation,

ut C uux C ı2uxxx D 0 ı2 � 1; (6.2)

as a continuum limit of the FPU system (6.1). They observed that solitary travelingwave solutions, now known as solitons, exist, whose interaction properties resultin analogous recurrences of initial states (Fig. 6.2). These results, of course, ledto the birth of a new field in applied mathematics dealing with the integrabilityof evolution equations and their solvability by the theory of Inverse ScatteringTransforms [2, 3].

In taking the continuum limit, however, Zabusky and Kruskal overlooked animportant aspect of the FPU problem, which is the discreteness of the particle chain.What happens often in discrete systems is that resonances between neighboringoscillators can be avoided if their vibration frequencies lie outside the so-calledphonon band representing the frequency spectrum of the linear problem. Thisleads to the occurrence of localized oscillations called discrete breathers, whichcertainly prevent equipartition of energy among all particles of the chain. Thus,one might conjecture that FPU recurrences are also the result of limited energytransport caused by discreteness and non-resonance effects. This, however, is anoversimplified way to view the FPU paradox.


The reason is that discrete breathers, of whom we shall have a lot more to sayin Chap. 7, are localized in configuration space and hence involve the oscillationsof few particles, while the initial state of the FPU experiment involves all particlesforming a single Fourier mode of the corresponding linear chain. Thus, we shouldbe looking instead at localization in Fourier q-modal space, as the FPU recurrenceswere observed after all when all the energy was placed in the q D 1 mode!

6.1.2 The Concept of q-breathers

This crucial fact was emphasized by Flach and his co-workers [128, 129], whointroduced the concept of q-breathers as exact periodic solutions of the problemand pointed out the role of their (linear) stability on the dynamics of FPU systems.In fact, they considered besides the FPU�˛ version of (6.1) the FPU�ˇ version aswell, described by the Hamiltonian

H D 1

2

NXkD1

y2k C 1

2

NXkD0

.xkC1 � xk/2 C ˇ

4

NXkD0

.xkC1 � xk/4 D E (6.3)

where xk again is the kth particle’s displacement from equilibrium, yk is itscanonically conjugate momentum and fixed boundary conditions are imposed bysetting x0 D xN C1 D 0. A q-breather is defined as the continuation, for ˇ ¤ 0

in (6.3) (or ˛ ¤ 0 in (6.1)) of the simple harmonic motion exhibited by the qthmode in the uncoupled case (˛ D ˇ D 0) and refers to the oscillation of allparticles with a single frequency nearly equal to the frequency of the qth linearmode. It is, therefore, exactly what we have called NNM in Chap. 4. For a recentand comprehensive review of these developments see [125].

Clearly, therefore, an important clue to the FPU paradox for small couplingparameters lies in the observation that, if we excite a single low q-breather mode(q D 1; 2; 3; : : :), the total energy remains localized in practice only within a fewof these modes, at least as long as the initially excited mode is stable. Furthermore,if we follow numerically FPU trajectories with initial conditions close to such aNNM, we find that they exhibit nearly the same localization profile as the NNM,at least for very long time intervals. Thus, NNMs surface again in our studies asq-breathers offering new insight in understanding the problem of deviations fromenergy equipartition among all modes, due to FPU recurrences.

Analytical estimates of various scaling laws concerning q-breathers were alsoobtained via the perturbation method of Poincare-Linstedt series in [129], whichwe shall describe in detail later in this chapter. Using this method, the authors of[129] expanded the solutions for all canonical variables up to the lowest non-trivialorder with respect to the small parameters and derived two important results: (a) Anasymptotic formula for the energy threshold of the first destabilization of the lowq-breathers of the FPU�ˇ chain (6.3)

6.1 The Fermi Pasta Ulam Problem 137

Ec � �2

6ˇ.N C 1/; (6.4)

valid for large N , and (b) an exponentially decaying function for the averageharmonic energy of the qth mode, E.q/ / exp.�bq/, where b depends onthe system’s parameters, N and the total energy E . Remarkably enough, theseq-breather energy profiles E.q/ are very similar to those of “FPU trajectories”,i.e. solutions generated by exciting initially one or a few low-q modes (as in theoriginal FPU experiments), whose exponential localization in q-space had alreadybeen noted earlier by other authors [110, 139, 140, 228].

Based on this similarity, we may therefore conjecture that there is a closeconnection between q-breathers and the energy localization properties of FPUtrajectories. This sounds as if we are on the right track, yet two intriguing questionsarise: What is the precise nature of this connection and why do FPU recurrencespersist at parameter values for which the corresponding q-breather solution becomesunstable? In this chapter, we shall try to answer these questions.

Before doing this, however, we need to make one more crucial remark: In 2004,Berchialla et al. [37] explored in detail the phenomenon of energy equipartition inFPU experiments, exciting initially a fixed (low-frequency) fraction of the spectrum,which is detached from zero i.e. ŒN=64; 5N=64�. The direct implication of theirnumerical study was that the flow of energy across the high-frequency parts of thespectrum takes place exponentially slowly, by a law of the form T / exp."�1=4/,where T is the time needed for the energy to be nearly equally partitioned, and" D E=N is the specific energy of the system. In fact, this dependence appears as apiecewise power law, with different “best-fit” slopes over different ranges of valuesof ".

Recently, a new study on equipartition times [30] offers new insight to thisquestion. These authors show numerically that the full FPU system, including both˛ and ˇ terms (see (6.1), (6.3)), does not exhibit exponentially long times toequipartition in the thermodynamic limit, while the FPU-ˇ model itself does so onlyfor initial conditions of the type chosen in [37]. Still, it is important to emphasizethat all results on equipartition times in the thermodynamic limit depend on thespecific energy of the system.

For example, a power-law behavior of the type T / "�3 was found in asubinterval of " values in [37], which fits nicely previous results reported in [112] onthe scaling of the equipartition time with " beyond a critical “weak chaos” threshold.Nevertheless, the question of which scaling laws correctly characterize the approachto equipartition remains open, since no rigorous results have so far been provided,while numerical results are available over a limited range of values of E and N . Fora recent review of various semi-analytical and numerical approaches to this issuesee [227].

In this context, an important observation made in [38] turns out to be quiterelevant: Even if one starts by exciting one low q-mode with large enough energy,one observes, long before equipartition, the formation of metastable states coined“natural packets”, in which the energy undergoes first a kind of equipartition among


a subset of (low-frequency) modes, as if only a fraction of the spectrum was initiallyexcited. These packets appear quite clearly in the FPU�˛ model, where the setof participating modes exhibits a “plateau” in the energy spectrum. In the FPU�ˇ

model on the other hand, although no clear plateaus are observed, the low-frequencypart of the spectrum contains most of the energy given initially to the system, in away that allows us to define an effective packet width. In fact, one such packetconsisting of five modes is exactly what FPU observed in their original experiment(see Fig. 6.1). We may thus conclude that FPU trajectories are characterized bya kind of metastability resulting from all types of initial excitations of the lowfrequency part of the spectrum [34].

Empirical scaling laws were established [38] concerning the dependence of the“width” of a packet (i.e. the effective number of participating modes), on the specificenergy ". These are consistent with the laws of energy localization obtained viaeither a continuous Hamiltonian model which interpolates the FPU dynamics inthe space of Fourier modes [24, 275], or the q-breather model [126]. A difference,however, between the two models is that in the continuous model the constant b inE.q/ / exp.�bq/ turns out to be independent of E .

6.1.3 The Concept of q-tori

Let us now think of q-breathers (or NNMs) as tori of dimension one, forminga “backbone” in phase space that describes the motion of the normal modevariables. The following question naturally arises: Since we are dealing with N dofHamiltonian systems, why not consider also the relevance of tori of any dimensionlower than N ? In particular, do such tori exist, and if so, do they exhibit similarlocalization properties as the q-breathers? As we explain in this chapter, a morecomplete interpretation of the “paradox” of FPU recurrences can be put forth, byintroducing the concept of q-tori, which reconciles q-breathers and their energylocalization properties, with the occurrence of metastable packets of low-frequencymodes. To achieve this deeper level of understanding of lack of equipartition, weneed to generalize the results obtained in [37, 112] for FPU trajectories, observingthat the packet of modes excited in these experiments corresponds to spectralnumbers satisfying the condition .N C 1/=64 � q � 5.N C 1/=64, with N equalto a power of two minus one.

We shall thus consider classes of special solutions lying not only on one-dimensional tori (as is the case with q-breathers), but also on tori of any lowdimension s � N , i.e. solutions with s independent frequencies, representing thecontinuation of motions resulting from exciting s modes of the uncoupled case,where s is allowed to vary proportionally to N . In what follows, we shallfocus on exploring the properties of such q-tori solutions, both analytically andnumerically. In particular, we shall establish energy localization laws analogousto those for q-breathers, using the Poincare-Linstedt perturbative method. We willthen demonstrate by numerical experiments that these laws describe accurately the

6.2 Existence and Stability of q-tori 139

properties not only of exact q-tori solutions, but also of FPU trajectories with nearbyinitial conditions.

The q-tori approach has been described in detail in a recent publication [94], sowe will only briefly sketch it here. Note that our aim in studying q-tori is to establishthe existence of exact invariant manifolds of lower dimension in the FPU problem.A similar approach can be found in the work of Giorgilli and Muraro [150], whoalso explored FPU trajectories confined in lower-dimensional manifolds of the 2N -dimensional FPU phase space. They proved that this confinement persists for timesexponentially long in the inverse of ", in the spirit of the famous Nekhoroshev theory[33,256]. In fact, the theory of Nekhoroshev offers a natural framework for studyinganalytically the metastability scenario.

In [94], it was shown that use of the Poincare-Linstedt method leads to accuratescaling laws for the energy profile E.q/ of a trajectory lying exactly on a q-torus,while the consistency of the Poincare-Linstedt construction on a Cantor set ofperturbed frequencies (or amplitudes) was explicitly demonstrated. Numerically,one finds that energy localization persists for long times also for trajectories near aq-torus, even beyond the energy threshold where the q-torus itself becomes unstable.Of course, the determination of linear stability for an s-dimensional torus (s > 1) isa subtle question, since no straightforward application of Floquet theory is available,as in the s D 1 case. Nevertheless, as shown in [94], one can employ the method ofGALI (see Chap. 5) to approximately determine critical parameter values at whicha low-dimensional torus turns unstable, in the sense that orbits in its immediatevicinity begin to display chaotic behavior.

Thus, in the next sections we briefly describe analytical and numerical resultson the existence, stability and scaling laws of q-tori, using the FPU�ˇ model as anexample. We focus on the subset of q-tori corresponding to zeroth order excitationsof a set of adjacent modes q0;i , i D 1; : : : ; s, whose energy profile E.q/ can beevaluated analytically and examine the energy localization of FPU trajectories whenthe linear stability of a nearby q-torus is lost. Finally, we address the question of thebreakdown of FPU recurrences and the onset of equipartition in connection with theslow diffusion of orbits starting in the vicinity of unstable q-tori.

6.2 Existence and Stability of q-tori

Let us begin our study of the FPU�ˇ system (6.3) introducing normal modecanonical variables Qq , Pq through the (linear) set of canonical transformations(see our discussion in Sects. 4.1 and 4.2)

xk Dr

2

N C 1

NXqD1

Qq sin

�qk�

N C 1

�; yk D

r2

N C 1

NXqD1

Pq sin

�qk�

N C 1

�:

(6.5)Direct substitution of (6.5) into (6.3) allows us to write the Hamiltonian in the formH D H2 C H4, in which the quadratic part represents N uncoupled harmonicoscillators


H2 D 1

2

NXqD1

�P 2

q C ˝2qQ2

q

�(6.6)

with (linear) normal mode frequencies

˝q D 2 sin

�q�

2.N C 1/

�; 1 � q � N; (6.7)

and the quartic part is as given in (4.9). Note that in this chapter, in contrast withChap. 4, we shall denote harmonic frequencies by the symbol ˝ and use ! for theperturbed frequencies in the context of the Poincare-Linstedt approach describedbelow. Thus, we write our equations of motion in the new canonical variables as

RQq C ˝2qQq D � ˇ

2.N C 1/

NXl;m;nD1

Cq;l;m;n˝q˝l˝m˝nQlQmQn (6.8)

(see (4.11)), using for the non-zero values of the coefficients Cq;l;m;n the expressionsgiven in (4.10). As we have already explained in Sect. 4.2, the individual harmonicenergies Eq D .P 2

q C ˝2qQ2

q/=2, which are constants of the motion for ˇ D 0,become functions of time for ˇ ¤ 0 and only the total energy E of the systemis conserved. It is, therefore, useful to define a specific energy as " D E=N andan average harmonic energy of each mode over a time interval 0 � t � T byNEq.T / D 1

T

R T

0Eq.t/dt .

In classical FPU experiments, one starts with the total energy shared only by asmall subset of modes. Thus, for short time intervals T , we have Eq.T / � 0 for allq corresponding to non-excited modes. Equipartition, therefore, means that (due tononlinearity) the energy will eventually be shared equally by all modes, i.e.

limT !1

NEq.T / D " ; q D 1; : : : ; N: (6.9)

Classical ergodic theory predicts that (6.9) will be violated only for orbits resultingfrom a zero measure set of initial conditions. The FPU “paradox”, on the other hand,owes its name to the crucial observation that large deviations from the approximateequality NEq.T / � " occur for many other orbits as well. Depending on the initialconditions, these deviations, termed FPU recurrences, are seen to persist even whenT becomes very large.

6.2.1 Construction of q-tori by Poincare-Linstedt Series

We now introduce the main elements of the method of q-torus constructionpresented in [94], using an explicitly solved example for N D 8, whose solutionslie on a two-dimensional torus representing the continuation, for ˇ ¤ 0, of the


quasi-periodic solution of the uncoupled (ˇ D 0) system Q1.t/ D A1 cos ˝1t ,Q2 D A2 cos ˝2t , for a suitable choice of A1 and A2.

To this end, we follow the Poincare-Linstedt method and look for solutionsQq.t/, q D 1; : : : ; 8, expanded as series of powers of the (small) parameter� D ˇ=2.N C 1/, namely

Qq.t/ D Q.0/q .t/ C �Q.1/

q .t/ C �2Q.2/q .t/ C : : : ; q D 1; : : : ; 8: (6.10)

For the motion to be quasiperiodic on a 2-torus, the functions Q.r/q .t/ must, at

any order r , be trigonometric polynomials involving only two frequencies (andtheir multiples). Furthermore, in a continuation of the unperturbed solutions Q1

and Q2, the new frequencies !1 and !2 of the perturbed system must representsmall corrections of the linear normal mode frequencies ˝1, ˝2. According to thePoincare-Linstedt method, these new frequencies are also given by series in powersof � , as

!q D ˝q C �!.1/q C �2!.2/

q C : : : q D 1; 2: (6.11)

As is well-known, the above corrections are determined by the requirement that allterms in the differential equations of motion, giving rise to secular terms (of theform t sin !qt etc.) in the solutions Qq.t/, be eliminated.

Let us consider the equation of motion for the first mode, whose first few termson the right side are

RQ1 C ˝21 Q1 D ��.3˝4

1 Q31 C 6˝2

1 ˝22Q1Q

22 C 3˝3

1˝3Q21Q3 C : : :/: (6.12)

Proceeding with the Poincare-Linstedt series, the frequency ˝1 is substituted onthe left side of (6.12) by its equivalent expression obtained by squaring (6.11) andsolving for ˝2

1 . Up to first order in � this gives

˝21 D !2

1 � 2�˝1!.1/1 C : : : : (6.13)

Substituting the expansions (6.10) into (6.12), as well as the frequency expansion(6.13) into the left hand side (lhs) of (6.12), and grouping together terms of likeorders, we find at zeroth order RQ.0/

1 C !21 Q

.0/1 D 0, while at first order

RQ.1/1 C !2

1 Q.1/1 D 2˝1!

.1/1 Q

.0/1 � 3˝4

1.Q.0/1 /3 � 6˝2

1˝22 Q

.0/1 .Q

.0/2 /2 �

�3˝31˝3.Q

.0/1 /2Q

.0/3 C : : : : (6.14)

Repeating the above process for modes 2 and 3, we find that their zeroth orderequations also take the harmonic oscillator form

RQ.0/2 C !2

2Q.0/2 D 0; RQ.0/

3 C ˝23Q

.0/3 D 0: (6.15)


Note that the corrected frequencies !1, !2 appear in the zeroth order equations forthe modes 1 and 2, while the uncorrected frequency ˝3 appears in the zeroth orderequation of mode 3 (similarly, ˝4; : : : ; ˝8, appear in the zeroth order equations ofmodes 4 to 8). Thus to construct a solution which lies on a 2-torus, we start fromparticular solutions of (6.15) (with zero velocities at t D 0) which read

Q.0/1 .t/ D A1 cos !1t; Q

.0/2 .t/ D A2 cos !2t; Q

.0/3 .t/ D A3 cos ˝3t;

where the amplitudes A1, A2, A3 are arbitrary. If the solution is to lie on a 2-toruswith frequencies !1, !2, we must set A3 D 0, so that no third frequency isintroduced in the solutions. In the same way, the zeroth order equations RQ.0/

q C˝2

qQ.0/q D 0 for the remaining modes q D 4; : : : ; 8, yield solutions Q

.0/q .t/ D

Aq cos ˝qt for which we require A4 D A5 D : : : D A8 D 0. We are, therefore, leftwith only two non-zero free amplitudes A1, A2.

Now, consider (6.14) for the first order term Q.1/1 .t/. Only zeroth order terms

Q.0/q .t/ appear on its right hand side (rhs), allowing for the solution to be found

recursively. The crucial remark here is that the choice A3 D : : : D A8 D 0,also yields Q

.0/3 .t/ D : : : D Q

.0/8 .t/ D 0, whence only a small subset of the

terms appearing in the original equations of motion survive on the right side of(6.14), namely those in which none of the functions Q

.0/3 .t/; : : : ; Q

.0/8 .t/, appears.

As a result, (6.14) is dramatically simplified and upon substitution of Q.0/1 .t/ D

A1 cos !1t , Q.0/2 .t/ D A2 cos !2t , reduces to

RQ.1/1 C !2

1Q.1/1 D 2˝1!

.1/1 A1 cos !1t � 3˝4

1A31 cos3 !1t �

�6˝21˝2

2 A1A22 cos !1t cos2 !2t: (6.16)

This can now be used to fix !.1/1 so that no secular terms appear in the solution,

yielding

!.1/1 D 9

8A2

1˝31 C 3

2A2

2˝1˝22 ;

while, after some simple operations, we find

Q.1/1 .t/ D 3A3

1˝41 cos 3!1t

32!21

C 3A1A22˝2

1˝22 cos.!1 C 2!2/t

2Œ.!1 C 2!2/2 � !21 �

C

C3A1A22˝

21 ˝2

2 cos.!1 � 2!2/t

2Œ.!1 � 2!2/2 � !21 �

: (6.17)

By the same analysis, we fix the frequency correction of the second mode to be

!.1/2 D 9

8A2

2˝32 C 3

2A2

1˝21˝2;


and obtain the solution

Q.1/2 .t/ D 3A3

2˝42 cos 3!2t

32!22

C 3A21A2˝2

1˝22 cos.2!1 C !2/t

2Œ.2!1 C !2/2 � !22 �

C

C3A21A2˝

21 ˝2

2 cos.2!1 � !2/t

2Œ.2!1 � !2/2 � !22 �

; (6.18)

which has a similar structure as the first order solution of the first mode. For thethird order term there is no frequency correction, and we find

Q.1/3 .t/ D A3

1˝31˝3

4

�3 cos !1t

!21 � ˝2

3

C cos 3!1t

9!21 � ˝2

3

�C (6.19)

C3A1A22˝1˝

22 ˝3

4

�cos.!1 � 2!2/t

.!1 � 2!2/2 � ˝23

C cos.!1 C 2!2/t

.!1 C 2!2/2 � ˝23

C 2 cos !1t

!21 � ˝2

3

�:

We may thus proceed to the sixth mode to find solutions in which all the functionsQ

.0/3 ; : : : ; Q

.0/6 , are equal to zero, while the functions Q

.1/3 ; : : : ; Q

.1/6 , are non zero.

However, a new situation appears when we arrive at the seventh and eighth modes.A careful inspection of the equation for the term Q

.1/7 .t/, i.e.

RQ.1/7 C ˝2

7 Q.1/7 D �

8Xl;m;nD1

C7;l;m;n˝7˝l˝m˝nQ.0/

l Q.0/m Q.0/

n ; (6.20)

shows that there can be no term on the rhs which does not involve some ofthe functions Q

.0/3 ; : : : ; Q

.0/8 . Again, this follows from the selection rules for the

coefficients in (4.10). Since all these functions are equal to zero, the rhs of (6.20) isequal to zero. Taking this into account, we set Q

.1/7 .t/ D 0, so as not to introduce

a third frequency ˝7 in the solutions, whence the series expansion (6.10) for Q7.t/

necessarily starts with terms of order at least O.�2/. The same holds true for theequation determining Q

.1/8 .t/.

Let us now make a number of important remarks regarding the above const-ruction:

1. Consistency. The solutions (6.17)–(6.19) (and those of subsequent orders)are meaningful only when the frequencies appearing in the denominators areincommensurable, so that denominators do not vanish. Note that the spectrum ofuncorrected frequencies ˝q , given by (6.7), is fully incommensurable if: (1) N

is a prime number minus one, or (2) a power of two minus one (see Problem 6.1and [163]).

In other cases, there are commensurabilities among the unperturbed frequen-cies, which do not affect the consistency of the construction of the Poincare-Linstedt series, because it can be shown that no divisors of the form

PNqD1 nq˝q ,


appear in the series at any order and any integer vector n � .n1; n2; : : : ; nq/. Theproof of this statement follows from the fact that differential equations like (6.16)determining all terms Q

.k/q , q D 1; : : : ; 8, at order k read

RQ.k/q .t/C!2

q Q.k/q .t/ D B.k/

q !.k/q cos !qt C

k�1Xn1;n22Z

jn1jCjn2j¤0

A.k/q;n1;n2

cosŒ.n1!1Cn2!2/t�;

(6.21)

if q D 1; 2, while for q D 3; : : : ; 8, the same equation holds with !2q on the left

side replaced by ˝2q . The coefficients B

.k/q and A

.k/q;n1;n2 in (6.21) are determined

at previous steps of the construction. It follows, therefore, that all new divisorsappearing at successive orders, belong to one of the following sets

.n1 ˙ 1/!1 C n2!2; n1!1 C .n2 ˙ 1/!2; ˝q ˙ .n1!1 C n2!2/; q D 3; : : : ; 8:

Thus, we deduce that no commensurabilities between the unperturbed frequen-cies ˝q can ever show up in the above series, as a zero divisor. For a more detailedanalysis of the consistency of the Poincare-Linstedt method as applied to theFPU ˇ-model (6.3) we refer the reader to [94]. In fact, the proof of consistencycan be generalized to construct s-dimensional q-tori, according to the nonlinearcontinuation of the set of linear modes qi , i D 1; : : : ; s, permitted by Lyapunov’stheorem [232]. For example, if the first s modes are excited with amplitudes Ak ,k D 1; : : : ; s, the formula for the perturbed frequencies reads

!q D ˝q C 3�

2˝q

sXkD1

A2k˝2

k � 3

8A2

q˝3q C O.�2I A1; : : : ; As/; (6.22)

for q D 1; : : : ; s. Fixing the frequencies !q in advance implies that (6.22)should be regarded as yielding the (unknown) amplitudes Ak for which thequasi-periodic solution exhibits the chosen set of frequencies. The case s D 1,of course, implies that the same property holds for the Poincare-Linstedtrepresentation of q-breathers described in [129].

2. Convergence. Even after establishing consistency, one still has to prove conver-gence of the series. However, as demonstrated in [120, 142, 143], the questionof convergence of the Poincare-Linstedt series is notoriously difficult even forsimple Hamiltonian systems. This is because the above series for orbits oninvariant tori are convergent, but not absolutely, as explained in [149]. Indeed,the problem of a suitable Kolmogorov construction of lower-dimensional toriyielding absolutely convergent series is still open (see e.g. [148]).

Thus, in [94] the construction of q-tori is justified by numerical simulations,taking initial conditions from the analytical expressions (6.10) at t D 0 andusing the GALI method of Chap. 5 to demonstrate that the computed orbits lie ontwo-dimensional tori. The numerical evaluation of the modes Q1.t/, Q3.t/ and


a b

c d

Fig. 6.3 Comparison of numerical (points) versus analytical (solid grey curve) solutions, usingthe Poincare-Linstedt series up to order O.�2/, for the time evolution of the modes (a) q D 1,(b) q D 3, (c) q D 7, when A1 D 1, A2 D 0:5, and N D 8, ˇ D 0:1. (d) The indices GALI2

to GALI6 up to a time t D 106 clearly indicate that the motion lies on a two-dimensional torus(after [94])

Q7.t/ follows remarkably well the analytical solution provided by the Poincare-Linstedt series truncated to second order in � , for N D 8, ˇ D 0:1, and A1 D 1,A2 D 0:5, as shown in Fig. 6.3a–c. The size of the error is as expected by thetruncation order in (6.10), while the GALI method shows, in Fig. 6.3d, that thenumerical orbit lies on a 2-torus.

Let us recall from Chap. 5 that the indicators GALIk, k D 2; 3; : : :, decayexponentially for chaotic orbits, due to the attraction of all deviation vectors bythe most unstable direction corresponding to the MLE. On the other hand, if anorbit lies on a stable s-dimensional torus, the indices GALI2; : : : ; GALIs oscillateabout a non-zero value, while the GALIsCj , j D 1; 2; : : : decay asymptoticallyby a power law. This is precisely what we observe in Fig. 6.3d. Namely, after aninitial transient interval, the GALI2 index stabilizes at a constant value GALI2 �0:1, while all higher indices, starting with GALI3, decay following power lawsexpected by the theory. Thus, we conclude that the motion lies on a 2-torus,exactly as predicted by the Poincare-Linstedt construction, despite the fact thatsome excitation was provided initially to all modes.

3. Presence and accumulation of small divisors. Even though zero denominatorsdo not occur, we cannot avoid the notorious problem of small divisors appearingin terms of all orders beyond the zeroth. Since the low-mode frequencies satisfy


!q � �q=N , divisors like !21 , or .!1 � 2!2/

2 ˙ !21 (appearing e.g. in (6.17)) are

small and care must be taken regarding their effect on the growth of terms of theseries at successive orders. For example, one of the most important such nearlyresonant divisors is 9!2

1 �˝23 in (6.19). However, since the first order corrections

of the frequencies !1 and !2 are O.ˇA2j =N 4/, for Aj , ˇ sufficiently small, one

may still use for all these frequencies the approximation given by the first twoterms in the sinus expansion of (6.7) namely

!q ' �q

.N C 1/� �3q3

24.N C 1/3; (6.23)

the error being O.A2j ˇ=N 4/ for !1; !2, and O..q=N /5/ for all other frequencies.

This implies that a divisor like 9!21 � ˝2

3 can be approximated by the relation

jq2!21 � ˝2

q j D j.q!1 C ˝q/.q!1 � ˝q/j ' 2q�

.N C 1/

�3.q3 � q/

24.N C 1/3

' �4q4

12N 4C O

�q6

N 6

�:

Defining the quantity

aq D �4q4

12N 4; (6.24)

we conclude that a divisor of order a3 appears in (6.19), which is the solutionof an equation of the first order in the recursive scheme. In general, the termsproduced at consecutive orders involve products of the form aq1 aq2 � � � aqr in thedenominators of the terms produced at the r th order of the recursive scheme.Consequently, this type of accumulation of small divisors does not lead to adivergent series, but can be exploited instead to obtain estimates of the profileof energy localization for the q-tori solutions, as we shall explain in the nextsection.

4. Sequence of mode excitations. The profile of energy localization along a q-torussolution can be determined by the sequence of mode excitations as the recursivescheme proceeds to higher orders. The term “excitation” here means that thesolutions of the Poincare-Linstedt method are non-zero for the first time at theorder where it is claimed that the excitation arises. For example, as alreadyexplained in the construction of a 2-torus solution, the modes 1 and 2 are excitedat zeroth order, the modes 3, 4, 5, and 6 at first order, and the modes 7 and 8 atsecond order.

Figure 6.4 presents one more example showing the comparison betweenanalytical and numerical results for the modes Q1.t/, Q5.t/, and Q13.t/, alonga 4-torus solution constructed precisely as described above, with N D 16, ˇ D0:1, by exciting modes 1–4 at zeroth order, via the amplitudes A1 D 1, A2 D 0:5,A3 D 1=3, A4 D 0:25. In this case, we find that the modes excited at the first


a b

c d

Fig. 6.4 Same as in Fig. 6.3, showing the existence of a 4-torus in the system with N D 16,ˇ D 0:1, when A1 D 1, A2 D 0:5, A3 D 0:333 : : :, A4 D 0:25. The time evolution Qq.t/ isshown for the modes (a) q D 1, (b) q D 5, (c) q D 13. (d) The GALIk , k D 2; 3; 4 indices areagain seen to stabilize after t � 104, while for k � 5 they continue to decrease by power laws(after [94])

order of the recursive scheme are q D 5 to q D 12, while the modes excited atsecond order are q D 13 to q D 16. Thus, according to the GALI method themotion lies on a 4-torus, since GALI5 is the first index to fall asymptotically ast�1 (see Fig. 6.4d).

The sequence of excitation of different modes, whose amplitudes at the r thorder have a pre-factor �r D .ˇ=2.N C1//r , plays a crucial role in estimating theprofile of energy localization. However, the contribution of small divisors to sucha pre-factor must also be examined. This is the subject of the next subsection, inwhich a Proposition is provided for the sequence of mode excitations, in thegeneric case of arbitrary N and arbitrary dimension s of the torus (with therestriction s < N ). The consequences of this proposition are then examinedon the localization profile of q-tori and nearby FPU trajectories.

6.2.2 Profile of the Energy Localization

Let us see what happens in the case of the solutions shown in Figs. 6.3 and 6.4.Figure 6.5 shows the average harmonic energy of each mode over a time spanT D 106 in the cases of the q-torus of Figs. 6.3 and 6.4, shown in Fig. 6.5a,b respectively. The numerical result (open circles) compares excellently with the


a b

Fig. 6.5 The average harmonic energy Eq of the qth mode as a function of q, after a timet D 106 for (a) the 2-torus and (b) 4-torus solutions (open circles) corresponding to the initialconditions used in Figs. 6.3 and 6.4 respectively. The stars are Eq values calculated via theanalytical representation of the solutions Qq.t/ by the Poincare-Linstedt series. The filled circlesshow a theoretical estimate based on the average energy of suitably defined groups of modes (see(6.34) and relevant discussion in the text) (after [94])

analytical result (stars) obtained via the Poincare-Linstedt method. The filled circlesin each plot represent “piecewise” estimates of the localization profile in groups ofmodes excited at consecutive orders of the recursive scheme.

The derivation and exact meaning of these theoretical estimates will be analyzedin detail below. Here we point out their main feature showing a clear-cut separationof the modes into groups following essentially the sequence of excitations predictedby the Poincare-Linstedt series construction. Namely, in Fig. 6.5a we see clearly thatthe decrease of the average energy NEq along the profile occurs by abrupt steps, withthree groups formed by nearby energies, namely the group of modes 1, 2, then 3 to6, and then 7, 8. The same phenomenon is also seen in Fig. 6.5b, where the groupingof the different modes follows precisely the sequence of excitations 1 to 4, 5 to 12,13 to 16 as predicted by the theory.

Let us now extend this approach to a more general consideration of the structureof solutions lying on low-dimensional tori. In particular, let us consider the casewhere only a subset of modes 1 � q1 < q2 < : : : < qs < N are excited at the zerothorder of the perturbation theory, assuming that Q

.0/q ¤ 0 iff q 2 fq1; q2; : : : ; qsg.

The modes q1; : : : ; qs , need not be consecutive. We wish to see how this type ofzeroth order excitation propagates at subsequent orders. More specifically, we wish

to determine for which values of q one has Q.k0<k/q D 0, Q

.k/q ¤ 0, i.e., the modes q

are first excited at the kth order. The answer is provided by the following Propositionproved in [94]:

Proposition 6.1. Let the starting terms of a Poincare-Linstedt series solution withs frequencies be set as


Q.0/qi

D Aqi cos.!qi t C �qi /; for i D 1; 2; : : : ; s; 1 � q1 � q2 � : : : � qs � N

Q.0/q D 0 for all q ¤ qi :

Then, besides the terms Qkqi

.t/, the Poincare-Linstedt series terms Q.k/q .t/ which

are permitted to be non-zero, at the kth order of the series expansion, are given bythe values of q D q.k/ satisfying

q.k/ Dj 2�.N C 1/ � mk j; .mk mod .N C 1// ¤ 0; (6.25)

where mk can take any of the values j ˙qi1 ˙ qi2 ˙ : : : ˙ qi2kC1j, with

i1; i2; : : : ; i2kC1 2 f1; : : : ; sg for any possible combination of the ˙ signs, and� the integer part of the fraction .mk C N /=2.N C 1/.

It is instructive to clarify the use of this Proposition on some simple examples:q-breathers: If we excite only one mode q1 at zeroth order, new modes are

excited one by one at subsequent orders of perturbation theory and one has mk D.2k C 1/q1. As long as mk � N , the newly excited modes are q.k/ D mk D.2k C 1/q1, i.e., exactly as predicted in [128].

two-dimensional q-tori: Assume we excite the modes q1 D 1, q2 D 2 at zerothorder. At first order (k D 1) we have again q D jqi1 ˙ qi2 ˙ qi3 j with i1; i2; i3 2f1; 2g. We readily find that the newly excited modes are q D 1 C 1 C 1 D 3,q D 1 C 1 C 2 D 4, q D 1 C 2 C 2 D 5, and q D 2 C 2 C 2 D 6. At order k D 2,the first newly excited mode is q D 1 C 1 C 1 C 2 C 2 D 7, while the last newlyexcited mode is q D 2 C 2 C 2 C 2 C 2 D 10. In general, at order k 1 the newlyexcited modes are 2.2k � 1/ C 1 � q � 2.2k C 1/.

s-dimensional q-tori: Now, assume we excite the modes q1 D 1, q2 D 2, : : :,qs D s at the zeroth order. In the same way as above we find that the newly excitedmodes at order k 1 are s.2k � 1/ C 1 � q � s.2k C 1/.

In order to study energy localization phenomena associated with FPU trajecto-ries, we need to construct estimates for all Qq.t/ terms participating in a particularq-torus solution in which the s first modes .q1; q2; : : : ; qs/ D .1; 2; : : : ; s/ areexcited at zeroth order of the theory, with amplitudes A1, A2, : : :, As , respectively.Denoting by q.k/ the indices of all modes that are newly excited at kth order,according to the above Proposition, we have

q.k/ 2 Ms;k � fmax.1; .2k � 1/s C 1/; : : : ; .2k C 1/sg ; (6.26)

whence the following useful estimates are obtained

8q.k/ 2 Ms;k; k 1 q.k/ . .2k C 1/s; !q.k/ . .2k C 1/�s

N C 1: (6.27)

Let us now define, in the space of trigonometric polynomials f containing (forsimplicity) only cosine terms


f DX

k

Ak cos.k � �/; (6.28)

the “majorant” normkf k D

Xk

jAkj: (6.29)

Since the terms of mode Qq.k/ .t/ satisfy Q.n/

q.k/ D 0, 8n < k, Q.k/

q.k/ ¤ 0, we deduce

that the equation determining Q.k/

q.k/ reads

RQ.k/

q.k/ C !2q.k/ Q

.k/

q.k/ D (6.30)

D �˝q.k/

k�1Xn1;2;3D0

n1Cn2Cn3Dk�1

0B@

X

.q.n1/;q.n2/;q.n3//2Dq.k/

˝q.n1/˝q.n2/˝q.n3/Q.n1/

q.n1/Q.n2/

q.n2/Q.n3/

q.n3/

1CA;

whereDq D ˚

.qj1 ; qj2 ; qj3/ W q ˙ qj1 ˙ qj2 ˙ qj3 D 0�

:

Assuming that the Poincare-Linstedt series are convergent, we may omit higherorder terms Q

.kC1/

q.k/ , Q.kC2/

q.k/ , : : :, as long as we seek an estimate of the size of

the oscillation amplitude of the mode q.k/. The average size of the oscillationamplitudes of each mode on an s-dimensional q-torus now follows from estimateson the norms of the various terms appearing in (6.30). If we denote by A.k/ the meanvalue of all the norms jjQq.k/ jj, we obtain the following estimate:

A.k/ D .C s/kA2kC10

2k C 1; (6.31)

where A0 � A.0/ is the mean amplitude of the oscillations of all modes excited atthe zeroth order of the perturbation theory, and C is a constant of order O.1/ [94].In fact, (6.31) is a straightforward generalization of the estimate given in [129] forq-breathers, while the two estimates become identical (except for the value of C ) ifone sets s D 1 in (6.31), and q0 D 1 in Flach’s q-breathers formulae.

The above analysis suggests that, starting with initial conditions near q-breathers,a “backbone” is formed in phase space by a hierarchical set of solutions whichare, precisely, the solutions lying on low-dimensional q-tori (of dimension s D1; 2; : : : ; s � N ). This also means that all FPU trajectories with initial conditionswithin this set exhibit a profile of energy localization characterized by a “stepwise”exponential decay, with step size equal to 2s, as implied by (6.26). Thus, all themodes q.k/ in this group share a nearly equal mean amplitude of oscillations givenby the estimate of (6.31).


Using (6.31), it is also possible to derive ‘piecewise’ estimates of the energy ofeach group using a formula for the average harmonic energies E.k/ of the modesq.k/. To achieve this, note that the total energy E given to the system can beestimated as the sum of the energies of the modes 1; : : : ; s (the remaining modesyield only small corrections to the total energy), i.e.

E � s!2q.0/A

20 � �2s3A2

0

.N C 1/2:

On the other hand, the energy of each mode q.k/ can be estimated from

E.k/ � 1

2˝2

q.k/

�ˇ

2.N C 1/

�2k

.A.k//2 � �2s2.C sˇ/2kA4kC20

22kC1.N C 1/2kC2;

which, in terms of the total energy E , yields

E.k/ � E

s

�C 2ˇ2.N C 1/2E2

�4s4

�k

: (6.32)

Once again, the similarity of (6.32) with the corresponding equation for q-breathers[129]

E.2kC1/q0 � Eq0

�9ˇ2.N C 1/2E2

64�4q40

�k

(6.33)

is evident, where q0 is the unique mode excited at zeroth order of the perturbationtheory. Note, in particular, that the integer s in (6.32) plays a role similar to that ofq0 in (6.33). This means that the energy profile of a q-breather with q0 D s obeysthe same exponential law as the energy profile of the s-dimensional q-torus. But themost important feature of the latter solutions is that their profile remains unalteredas N increases, provided that: (1) a constant fraction M D s=N of the spectrumis initially excited, (i.e. that s increases proportionally to N ), and (2) the specificenergy " D E=N remains constant. Indeed, in terms of the specific energy ", (6.32)takes the form

E.k/ � "

M

�C 2ˇ2"2

�4M 4

�k

; (6.34)

i.e. the profile becomes independent of N . A similar behavior is observed withq-breather solutions provided that the “seed” mode q0 varies linearly with N , aswas shown explicitly in [130, 179].

The same q-torus construction can be performed for the FPU�˛ model as well.Here we only discuss the main results regarding the exponential localization ofq-torus solutions, whose profile is quite similar to the one given in (6.34).


6.3 A Numerical Study of FPU Trajectories

Examples of the “stepwise” profiles predicted by (6.32) in the case of exact q-torisolutions are shown by filled circles in Fig. 6.5, corresponding to the solutionsshown in Figs. 6.3 and 6.4 (in all fittings we set C D 1 for simplicity). Fromthese we see that the theoretical “piecewise” profiles yield nearly the same averageexponential slope as the profiles obtained numerically, or analytically by theconstruction of the solutions via the Poincare-Linstedt method . Thus, the estimates(6.32), or (6.34), appear quite satisfactory for characterizing the localization profilesof exact q-tori solutions.

The key question now, regarding the relevance of the q-tori solutions for theinterpretation of the FPU paradox, is whether (6.32) or (6.34) retain their predictivepower in the case of generic FPU trajectories which, by definition, are started closeto but not exactly on a q-torus.

To answer this question, let us examine the results plotted in Figs. 6.6 and 6.7.Figure 6.6 shows the energy localization profile in numerical experiments on the

a b c

d e f

Fig. 6.6 The average harmonic energy Eq of the qth mode over a time span t D 106 as a functionof q in various examples of FPU trajectories, for ˇ D 0:3, in which the s.D N=16/ first modesare only excited initially via Qq.0/ D Aq , PQq.0/ D 0, q D 1; : : : ; s, with the Aq selected so thatthe total energy is equal to the value E D H indicated in each panel. We thus have (a) N D 64,E D 10�4, (b) N D 128, E D 2 � 10�4, (c) N D 256, E D 4 � 10�4, (d) N D 64, E D 10�3,(e) N D 128, E D 2 � 10�3 , (f) N D 256, E D 4 � 10�3. The specific energy is constantin each of the two rows, i.e. " D 1:5625 � 10�6 in the top row and " D 1:5625 � 10�5 in thebottom row. The dashed lines represent the average exponential profile Eq obtained theoreticallyby the hypothesis that the depicted FPU trajectories lie close to q-tori governed by the profile (6.34)(after [94])

6.3 A Numerical Study of FPU Trajectories 153

a b c

fed

Fig. 6.7 Same as in Fig. 6.6a but for larger energies, namely (a) E D 0:05, (b) E D 0:1, (c)E D 0:2, (d) E D 0:3, (e) E D 0:4, (f) E D 0:5. Beyond the threshold E � 0:05, theoreticalprofiles of the form (6.32) yield the correct exponential slope if s is gradually increased from s D 4

in (a) and (b) to s D 6 in (c), s D 7 in (d), s D 10 in (e), and s D 12 in (f) (after [94])

FPU�ˇ model in which ˇ is kept fixed (ˇ D 0:3), while N takes the values N D64, N D 128 and N D 256. Although the derivation of explicit q-tori solutionsis practically feasible by computer algebra only up to a rather small value of N

(N D 16), in the present section we numerically follow trajectories for much largervalues of N .

In all six panels of Fig. 6.6 the FPU trajectories are computed starting with initialconditions in which only the s D 4 (for N D 64), s D 8 (for N D 128) ands D 16 (for N D 256) modes are excited at t D 0, with the excitation amplitudescorresponding to the total energy E indicated in each panel, and constant specificenergy " D 1:5625 10�6 in the top row and " D 1:5625 10�5 in the bottom rowof Fig. 6.6.

The resulting trajectories differ from q-tori solutions in the following sense: Forthe q-tori all modes have an initial excitation whose size was estimated in (6.31) or(6.32), while in the case of the FPU trajectories only the s first modes are excitedinitially, and one has Qq.0/ D 0 for all modes q > s. Furthermore, since in theq-tori solutions one also has kQq>s.t/k � kQq�s.t/k for all t , the FPU trajectoriescan be considered as lying in the neighborhood of a q-torus, at least initially. Thenumerical results suggest that if E is small, FPU trajectories remain close to theq-tori even after relatively long times, e.g. t D 106.


This is depicted in Fig. 6.6, in which one sees that the average energy profiles ofthe FPU trajectories (at t D 106) exhibit the same behavior as predicted by (6.32),for an exact q-torus solution with the same total energy as the FPU trajectory ineach panel. For example, based on the values of their average harmonic energy, themodes in Fig. 6.6a (in which s D 4) are clearly separated in groups, 1 to 4, 5 to 12,13 to 20, etc., as foreseen by (6.26) for an exact 4-torus solution. The energies of themodes in each group have a sigmoid variation around a level value characteristic ofthe group, which is nearly the value predicted by (6.32). The grouping of the modesis distinguishable in all panels of Fig. 6.6, a careful inspection of which verifies thatthe grouping follows the laws found for q-tori. Also, if we superpose the numericaldata of the three top (or bottom) panels we find that the average exponential slopeis nearly identical in all panels of each row, a fact consistent with (6.34), accordingto which, for a given fraction M of initially excited modes, this slope depends onlyon the specific energy, i.e. it is independent of N for constant ".

If we now increase the energy, the FPU trajectories resulting from s initiallyexcited modes start deviating from their associated exact q-tori solutions. As aconsequence, the energy profiles of the FPU trajectories also start to deviate from theenergy profiles of the exact s-tori. This is evidenced by the fact that the profiles ofthe FPU trajectories become smoother, and the groups of modes less distinct, whileretaining the average exponential slope as predicted by (6.34). This “smoothing”of the profiles is discernible in Fig. 6.7a, in which the energy is increased by afactor of 50 with respect to Fig. 6.6d, for the same values of N and ˇ. Also, inFig. 6.7a we observe the formation of the so-called “tail”, i.e. an overall rise ofthe localization profile at the high-frequency part of the spectrum, accompanied byspikes at particular modes. This is a precursor of the breakdown of FPU recurrencesand the evolution of the system towards energy equipartition, which occur earlier intime as the energy becomes larger.

It is important to note that exponential localization of FPU trajectories persistsand is still characterized by laws like (6.32) even when the energy is substantiallyincreased. Furthermore, at energies beyond a threshold value, an interesting phe-nomenon is observed: For fixed N (see e.g. Fig. 6.7, where N D 64), as the energyincreases, a progressively higher value of s needs to be used in (6.32), so that thetheoretical profile yields an exponential slope that agrees with the numerical data.When ˇ D 0:3, N D 64, this threshold value, E � 0:1, splits the system in twodistinct regimes: One for E < 0:1, where the numerical data are well fitted by aconstant choice of s D 4 in (6.32) (indicating that the FPU trajectories are indeedclose to a 4-torus), and another for E > 0:1, where “best-fit” models of (6.32) occurfor values of s increasing with the energy, e.g. s D 6 for E D 0:2, rising to s D 12

for E D 0:5. This indicates that the respective FPU trajectories are close to q-toriwith a progressively higher value of s > 4, despite the fact that only the four firstmodes are excited by the initial conditions.

This behavior is analogous to the “natural packet” scenario described byBerchialla et al. [38], who observed that the localization profile of their metastablestates begins to stabilize as the energy increases beyond a certain threshold. Indeed,according to (6.32), such a stabilization implies that in the second regime the width

6.3 A Numerical Study of FPU Trajectories 155

s depends asymptotically on E as s / E1=2, or, from (6.34) M / "1=2. Thisagrees well with estimates on the width of natural packets in the FPU�ˇ modelas described in [227].

6.3.1 Long Time Stability near q-tori

It follows from the above results that as the exponential localization of the energyin the low q-modes relaxes, the FPU recurrences start to breakdown sooner andequipartition will eventually take place beyond some energy threshold. The questionremains, however: What does all this have to do with the stability of the q-tori? Howis the dynamics in their vicinity related to the properties of the motion more globallyas the energy increases?

The linear stability of q-breathers, of course, just as any other periodic solution,can be studied by the implementation of Floquet theory (see [129]), which demon-strates that a q-breather is linearly stable as long as

R D 6ˇEq0 .N C 1/

�2< 1 C O.1=N 2/: (6.35)

This result is obtained by analyzing the eigenvalues of the monodromy matrix ofthe linearized equations about a q-breather solution constructed by the Poincare-Linstedt series.

In the case of q-tori, however, the above techniques are no longer available.Nevertheless, a reliable numerical criterion for the stability of q-tori is providedby the GALI method. According to this approach, if a q-torus becomes unstablebeyond a critical energy threshold, the deviation vectors of trajectories started on(or nearby) the torus follow its unstable manifold, whence all GALIs fall to zeroexponentially fast. Most trajectories in that neighborhood are weakly chaotic andbegin to diffuse slowly away to larger chaotic seas, where no FPU recurrences occurand equipartition reigns. Since all GALIk in that vicinity decay exponentially aftera transient time, this weak chaos transition at low energies can be directly inferredfrom the time evolution of the lowest index, i.e. GALI2.

Using this criterion, one can study the stability of q-tori and determine approx-imately the value of the critical energy Ec at which a q-torus turns from stableto unstable. Note first that, due to the obvious scaling of the FPU Hamiltonian byˇ, Ec is well fitted by a power law Ec D Aˇ�1 for all N considered, as shownin Fig. 6.8a for N D 64. Here, the quantity A is obtained by keeping N fixed andvarying ˇ, until a critical energy is determined, beyond which the GALI2 index losesits (nearly) constant behavior. All calculations in Fig. 6.8 refer to a maximum timetmax D 107 up to which the exponential fall-of of GALI2 is observed. In general,Ec decreases as tmax increases, but appears to tend to a limiting value as tmax ! 1.


a b

Fig. 6.8 (a) The critical energy Ec , at which the GALI2 index shows that a q-torus destabilizes,is fitted by the solid line Ec D Aˇ�1, with A D 0:164, for fixed N D 64 (the dashed line showsthe �1 slope separately for clarity). (b) Plotting A as a function of N for an FPU trajectory startedby exciting initially the q D 1; 2; 3 modes (squares), or the q D 1; 2 modes (triangles), for fixedˇ D 0:3 yields curves that lie well above the threshold (filled circles) at which FPU trajectoriesnear a q D 1 breather solution destabilize (the dashed line depicts the A � N �1 law (6.35)). Thetotal integration time for both panels is tmax D 107 (after [92])

Thus, the values of Ec found in this way provide an upper estimate for the transitionenergy at which the exact q-torus turns unstable [92].

The fitting constant A, however, does depend on N , as seen in Fig. 6.8b, whereFPU trajectories are studied exciting the q D 1; 2; 3 modes, the q D 1; 2 modes anda q D 1 breather. The data marked by squares (3-torus) and triangles (2-torus) inFig. 6.8b clearly show that the dependence of A on N is considerably weaker thanthe N �1 law, (6.35), derived for FPU trajectories started near the q D 1 breather. Inthat case, the data shown by filled circles in Fig. 6.8b have a slope much closer to thatpredicted by (6.35), although the numerical curve is shifted upwards with respect tothe dashed line. These results evidently demonstrate that the GALI2 method yieldscritical energies Ec which are quite higher than the values at which the q-breatherbecomes unstable.

In summary, the upper two curves of Fig. 6.8b strongly suggest that the q-torisolutions are more robust than the q-breathers, regarding their stability under smallperturbations. Furthermore, they imply that the destabilization of the simple periodicorbit represented by a q-breather does not imply that the tori surrounding theq-breather are also unstable.

At any rate, it is important to remark that the exponential localization ofFPU trajectories persists even after the associated q-breathers or q-tori have beenidentified as unstable by the GALI criterion. This is exemplified in Fig. 6.9, wherepanels (a) and (c) show the time evolution of the GALI2 index for two FPUtrajectories started in the vicinity of 2-tori of the N D 32 and N D 128

systems, when ˇ D 0:1 and tmax D 107. In both cases, the energy satisfies E > Ec ,

6.4 Diffusion and the Breakdown of FPU Recurrences 157

a

c d

b

Fig. 6.9 (a) Time evolution of GALI2 up to tmax D 107 for an FPU trajectory started by excitingthe q D 1 and q D 2 modes in the system N D 32, ˇ D 0:1, with total energy E D 2, i.e. higherthan Ec D 1:6. (b) Instantaneous localization profile of the FPU trajectory of (a) at t D 107 . (c)Same as in (a) but for N D 64, E D 1:323 (in this case the critical energy is Ec D 1:24). (d)Same as in (b) but for the trajectory of (c) (after [94])

as the exponential fall-off of the GALI2 index is already observed at t D 107.Nevertheless, a simple inspection of Figs. 6.9b, d clearly shows that the exponentiallocalization of the energy persists in the Fourier space of both systems.

6.4 Diffusion and the Breakdown of FPU Recurrences

Let us now study the physically important phenomenon of diffusion in an FPUchain, when initial conditions are given near the first normal mode of the system. Inparticular, we shall consider here the FPU-˛ model and show that the evolution of


its dynamics breaks down into two stages during which the qualitative behavior ofthe motion is considerably different.

In direct analogy with the FPU-ˇ model, the FPU�˛ system describes a one-dimensional array of N particles governed by the Hamiltonian

H D 1

2

NXkD1

y2k C 1

2

NXkD0

.xkC1 � xk/2 C ˛

3

NXkD0

.xkC1 � xk/3; (6.36)

where we have imposed again fixed boundary conditions x.0/ D x.N C 1/ D 0.In the harmonic limit (˛ D 0), the Hamiltonian is identical to its quadratic part

H2 (see (6.6)). Thus, (6.36) can be expressed, through the canonical transformation(6.5), in Pq; Qq variables as

H D 1

2

NXqD1

.P 2q C ˝2

qQ2q/ C ˛

3p

2.N C 1/

NXq;l;mD1

Bq;l;m˝q˝l˝mQqQlQm;

(6.37)with ˝q denoting the linear normal mode frequencies (6.7) and Bq;l;m the couplingcoefficient between the modes given by the formula

Bq;l;m DX˙

.ıq˙l˙m;0 � ıq˙l˙m;2.N C1//; (6.38)

while the equations of motion of the system are given by

RQq C ˝2qQq D � ˛p

2.N C 1/

NXq;l;mD1

Bq;l;m˝q˝l˝mQlQm: (6.39)

6.4.1 Two Stages of the Diffusion Process

The dynamics of the FPU-˛ chain is studied systematically in [276], where twostages are found to occur: (I) The energy is suddenly transferred to a small numberof modes, (II) a metastable state arises and a slow diffusive process is observedleading the system finally to equipartition. Let us use as initial condition the firstnormal mode q D 1 of the FPU-˛ chain given by

xk D A sin�k

N C 1; Pxk D 0; (6.40)

where k D 1; 2; : : : ; N and A is arbitrary. If we denote by eq D Eq=E thenormalized instantaneous energies of the normal modes, one finds for small times(typically of the order of T D 103) that they are governed by an expression ofthe form


eq � t2.q�1/; 8q 2 f1; 2; : : : ; N g: (6.41)

This formula, in fact, represents Stage I of the diffusion process, as we discuss below(for more details see [276]).

Stage I:

In Stage I the energy diffuses from the initially excited normal mode to theremaining ones at a rate that is well described by the power law (6.41), as has alsobeen verified in [92] at energies E D 0:01, E D 0:1 and E D 1, with N D 31 and˛ D 0:33. This first type of energy transport occurs over a short time interval,beyond which (see Stage II below) the linear mode energies settle down to anexponential profile in q space. When the energy is small enough the spectral energyprofile is quite close to the corresponding q-breather of the FPU-˛ model.

However, when the energy attains higher values, the system forms a packet ofmodes that undergoes a type of internal equipartition of its energy. Referring to(6.41), the authors of [276] explain this behavior through near resonances of theform !q �q!1, which appear as denominators in the perturbation scheme. For shorttimes, these denominators are inversely proportional to t and hence the energy e2

is proportional to t2. Recursively, one can thus find that the remaining modes havenormalized energies which increase as eq � t2.q�1/, signifying that the resonancethat develops over that time period is responsible for the sudden transport of energyfrom the long wavelength normal modes to those of short wavelength.

Let us denote by TI the time needed for the FPU-˛ chain to pass from Stage Ito Stage II of the dynamics. Computations show that TI � 4000 for E D 0:01,TI � 1000 for E D 0:1 and TI � 400 for E D 1.

Stage II:

Stage I lasts until the frequencies in the denominators of the solution cease to bein resonance and the harmonic energies settle down to a definite value. Given thatwe have chosen as initial condition the q D 1 normal mode, the first modes to besuccessively excited are ˛1=2�1=4N [24, 37] and share within them almost all theenergy of the system, forming a well-defined packet as described in Sect. 6.1.2. Theremaining ones, i.e. the tail modes, have exponentially localized energy spectrum.This occurs for t TI and is called Stage II of the process. The FPU-˛ chainremains in that state until the resonances between the modes in the packet and tailmodes complete the transfer of energy between them and the system moves awayfrom the exponential profile.

Let us examine this stage of the dynamics for the following values of the totalenergy: E D 0:01, E D 0:1 and E D 1, setting N D 31 and ˛ D 0:33. For E D0:01 and E D 0:1 one observes that the energy spectrum of the FPU-˛ modelexhibits “spikes” at some modes, due to resonances with the tail modes, similar tothose seen in Fig. 6.7 for the FPU�ˇ model. The exponential slope of the profile,however, is the same, while the linear energy spectrum is invariant for exponentially


long times. On the contrary, if the total energy of the FPU-˛ system exceeds thevalue 0.1, the transfer of energy from the packet to the tail modes occurs at a muchfaster rate.

It follows from these results that there is a critical energy value below which FPUtrajectories remain close to the q-breather for exponentially long times. This criticalvalue can be calculated if we consider that the length of the mode packet is nearlyequal to

q

N' p

� D ˛1=2"1=4; (6.42)

whence for q D 1 it follows that this critical energy threshold is

Ec D 1

˛2N 3: (6.43)

We conclude, therefore, that for low energy values the Eq spectrum of theFPU-˛ chain is localized for exponentially long times, due to the FPU recurrencephenomenon and the existence of q-tori discussed in the previous section. It isinteresting that the spectrum remains localized for energy values E D 0:1 muchhigher than the critical value Ec D 2:8 10�4. In fact, for E D 0:1 (˛ D 0:33,N D 31) we observe through (6.42), that we have a packet of length 4, i.e. theenergy is equally distributed among the 4 modes q D 1; 2; 3; 4. This means that theFPU trajectory converges to a four-dimensional torus, even though initially only onemode was excited.

Stage II concludes with the localized energy of the FPU-˛ model diffusing fromthe normal modes of the initial packet to the tail modes leading the system toequipartition. Studying the rate of diffusion from the low to the high frequencymodes, the following has been recently discovered in [276]: (a) The FPU-˛ modelis characterized by sub-diffusion and (b) by measuring the diffusion rate one canestimate the time it takes for the system to reach equipartition. As is well-known,this is a delicate issue, which has been studied to date by many authors (seee.g. [37, 38, 110, 112, 275, 276]).

The main difficulty lies in the fact that the onset of energy equipartition ispractically impossible to observe, based only on the numerical integration of theequations of motion. Consequently the only tool we have to predict this onset withsome degree of accuracy is to use the theoretical prediction of the diffusion rate ofthe motion described below.

6.4.2 Rate of Diffusion of Energy to the Tail Modes

We shall now obtain a law of diffusion using the sum of the last third of thenormalized linear energies eq . The reason for choosing this set with the shortestwavelengths is because they exhibit the largest variations in their harmonic energies


Fig. 6.10 Time evolution of the energy of tail modes for the FPU-˛ model, with N D 31 and˛ D 0:33. The bottom curve corresponds to total energy E D 0:01, the middle one to E D 0:1

and the top curve to E D 1, for the instantaneous normalized harmonic energies eq (after [92,276])

and consequently yield a more efficient criterion for predicting of the time requiredfor energy equipartition.

The sum of normalized harmonic energies for the mode interval q 2 Œ2N=3; N �

is called tail mode energy in [276] and is defined by

.t/ DNX

qD2N=3

eq.t/; (6.44)

for the spectrum of energies eq.In Fig. 6.10 we plot the function .t/, for the energy values E D 0:01, E D 0:1

and E D 1, in log-log scale as a function of time and observe a diffusion law ofthe form

/ Dt : (6.45)

Let us now determine the dependence of the parameters D and on the totalenergy of the system E , given that � D ˛1=2"1=4 is the only independent parameterof the problem. To this end, let us examine the FPU-˛ system with N D 31 and˛ D 0:33, for 200 values of the total energy, starting with E D 0:01 by 0.01increments up to the value E D 2. In Fig. 6.11 we estimate the value of for theenergies of the tail modes as a function of E , using the method of least squares,for suitably long time intervals during which the tail mode energies satisfy the law


Fig. 6.11 Plot of the value of of the energies of the tail modes of the FPU-˛ chain, withN D 31 and ˛ D 0:33 (after [92, 276])

(6.45). Since for every energy value we have a different time interval, during whichthe tail mode energies grow linearly (in log-log scale), it is practically impossible torepeat the calculation for every value of E . For this reason we split the computationin three energy ranges: (a) E 2 Œ0; 0:51�, where the appropriate time interval isŒ104; 3 106�, (b) E 2 Œ0:52; 0:66�, where this interval is Œ3 103; 106� and (c)E 2 Œ0:67; 2�, for which the time interval is Œ103; 3 105�.

Thus, in Fig. 6.11, plotting in (6.45) as a function of E , we find that the systemexhibits subdiffusion, since the power of t in (6.45) is generally smaller than unity.

6.4.3 Time Interval for Energy Equipartition

As we deduce from the results shown in Fig. 6.10, the system is characterized byenergy equipartition when D 1=3. Thus, solving from the expression (6.45) fort D T eq , with D 1=3, we can estimate the equipartition time of the FPU-˛ systemto be

T eq � .3D/�1= : (6.46)

Evaluating the parameters D and from our knowledge of the total energy E ofthe FPU-˛ model with N D 31 and ˛ D 0:33, we use (6.46) to calculate theequipartition time of the system. In Fig. 6.12 we plot the logarithm of this timeas a function of energy of the system, in semi-log plot, for 200 energy valueswithin the intervals: (a) E 2 Œ0:001; 0:2� where the energy increment is 0:001


Fig. 6.12 The logarithm of the equipartition time T eq for the FPU-˛ chain, with N D 31 and˛ D 0:33, as a function of the total energy, in semi-log scale, for: (a) E 2 Œ0:001; 0:2� and (b)E 2 Œ0:01; 2�. In the insert of (b) we show the logarithm of the equipartition time T eq as a functionof the logarithm of E (after [92, 276])

and (b) E 2 Œ0:01; 2�, where the energy increases by steps of 0:01. In the smallinsert of Fig. 6.12b we depict the quantity log T eq vs. log E , and find that in theenergy interval E 2 Œ0:4; 2�, the law for the time required for equipartition may beexpressed as

T eq � 106:25E�3: (6.47)

This agrees very well with the formula given in [112] for the same energyinterval. Even so, for energy values within 0:1 < E < 0:4 the system takesexponentially long to reach equilibrium, while for E � 0:1 equipartition has notyet been observed. The theoretical prediction of the relation (6.46) diverges fromthe numerical results for 0:1 < E < 0:4, while for E � 0:1, it is impossible tocheck its validity with present day available numerical techniques.

As we see from Fig. 6.10, the slope of the -curve for E D 0:01 is practicallyzero. The problem is that its value is smaller than the errors of the least squaremethod. Therefore, the conclusion is that the validity of these results is questionablefor E � 0:1. To date there is no theoretical estimate in the literature allowing us todecide whether the system reaches equipartition when E becomes too small. Thus,there are two possibilities: Either the equipartition times T eq are exponentially long,or the system remains trapped in the neighborhood of a q-torus or a q-breather. Theformulas described above cannot answer the question whether the system reachesequipartition for E � 0:1, but do confirm the validity of the law T eq � E�3 givenin [112], for energies E 2 Œ0:4; 2�.

Exercises

Exercise 6.1. Show that the spectrum of the unperturbed (harmonic) frequencies˝q , given by (6.7), is fully incommensurable if N is either a prime number minusone, or a power of two minus one. Hint: See Exercise 4.1.


Problems

Problem 6.1. (a) Carry out in detail the steps outlined in Sect. 6.2.1 and apply thePoincare-Linstedt series method to construct a 2-torus solution with N D 8,and a 4-torus solution with N D 16 for the FPU�ˇ model (6.3) with ˇ D 0:1.

(b) Choosing initial conditions on these tori, implement numerically the methodsof SALI and GALI described in Chap. 5 to show that these tori are stable forsmall enough values of the total energy E .

(c) Increasing the value of E , determine critical values E D Ec.2/ and E D Ec.4/,at which these tori first become unstable.

Problem 6.2. Repeat the calculations of Problem 6.1 for the FPU-˛ model, withHamiltonian (6.36) and ˛ D 0:1. Consult the beginning of Sect. 6.4, where theappropriate transformations to canonical variables Qq; Pq are described. Comparethe results with the corresponding findings obtained for the FPU-ˇ model inProblem 6.1.

Chapter 7Localization and Diffusion in NonlinearOne-Dimensional Lattices

Abstract In this chapter we focus on localization properties of nonlinear latticesin the configuration space of their spatial coordinates. In particular, we begin bydiscussing the phenomenon of exponentially localized periodic oscillations, calleddiscrete breathers, in homogeneous one-dimensional lattices. Demonstrating thatthey are directly connected with homoclinic orbits of invertible maps, we describea precise numerical procedure for constructing them to arbitrarily high accuracy.Furthermore, the homoclinic approach allows for the symbolic classification of allmultibreathers (having more than one major amplitudes) symmetric and asymmetricand can be extended to 2D lattices. We also analyze the (linear) stability of breathersand show how their bifurcations can be monitored using methods of feedback con-trol. Chapter 7 ends with a description of important recent results on delocalizationand diffusion of wavepackets in lattices with random disorder, pointing out thecomplexity of these phenomena and the open questions that still remain.

7.1 Introduction and Historical Remarks

Localization phenomena in systems of many (often infinite) degrees of freedomhave frequently attracted attention in solid state physics, nonlinear optics, super-conductivity and quantum mechanics. It is well-known, of course, that localizationin lattices can be due to disorder (see e.g. [1]), while, if the disorder is random,one encounters the famous Anderson localization phenomenon [10]. There is,however, another very important type of localization, occurring in homogeneousinfinite dimensional Hamiltonian lattices, which is dynamic: It refers to localizedperiodic oscillations arising not because of the presence of some defect, but due tothe interaction between nonlinearity and the spectrum of linear normal modes (orphonons) of the system.

We are talking, in particular, about a remarkable phenomenon called discretebreathers, representing periodic oscillations of very few sites, in nonlinear latticesconsisting, in principle, of infinitely many particles! It is surprisingly common in


165

166 7 Localization and Diffusion in Nonlinear One-Dimensional Lattices

discrete systems, in contrast to continuum systems (like strings, membranes or fluidsurfaces), where it only appears in certain very special integrable examples. Sincethe reader may not be too familiar with these concepts, we shall first present a briefreview of the history of discrete breathers and then describe how one can studythem using homoclinic orbits of multi-dimensional maps. This approach not onlyprovides a very efficient method for computing discrete breathers, but also leads to asystematic way of classifying them and allows us to accurately follow their existenceand stability properties as certain physical parameters of the problem are varied.

At the beginning we will deal exclusively with homogeneous lattices (of identicalmasses and spring constants) and proceed in subsequent sections to treat the problemof diffusion in disordered lattices starting with an initial excitation of very few sites,which has recently received a great deal of attention in the literature.

7.1.1 Localization in Configuration Space

In 1969, while studying a system of coupled nonlinear oscillators modelingfinite-sized molecules, Ovchinnikov [261] showed in a very beautiful way howdiscreteness in combination with an intrinsic nonlinearity of the system can causeenergy localization of the type discovered by FPU. He demonstrated, in particular,that resonances between neighboring oscillators can be avoided when the vibrationsof the system lie outside the so-called phonon band, or spectrum of linear normalmode frequencies (see Fig. 7.1). Thus, the recurrence phenomena in the FPUexperiments can be explained as the result of limited energy transport betweenFourier modes, caused by discreteness and nonlinearity effects. Today, of course, ourunderstanding of this phenomenon is considerably enhanced through the analysis ofexponential localization in Fourier space described in detail in Chap. 6.

Twenty years later (1988), the subject of localized oscillations in nonlinearlattices was revived in a seminal paper by Sievers and Takeno [306], who usedanalytical arguments to show that energy localization occurs generically in FPUsystems of infinitely many particles in one dimension, obeying (6.1).

Combining their perturbative analysis with numerical experiments, they demon-strated that a new class of solutions, known to date as discrete breathers, existas oscillations which are both time-periodic and spatially localized. In theirsimplest form, these solutions exhibit significant excitations only of the middle andnearby particles. However, a great many patterns are also possible, the so-calledmultibreathers, in which several particles oscillate with large amplitudes, as shownhere in Fig. 7.2. How can one determine all the possible shapes? Which of them arestable under small perturbations? These are the kind of questions we would like toaddress in the next section.

Besides the FPU system, the existence of these localized oscillations was soonverified numerically by other research groups on a variety of lattices, including the

7.1 Introduction and Historical Remarks 167

00

0.5

1

1.5

E1,

E2

E1,

E2

2

2.5

10 20 30 40 50

t60 70 80 90 100 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

10 20 30 40 50

t60 70 80 90 100

a b

Fig. 7.1 Local energies E1 .t/ (solid line) and E2 .t/ (dashed line) of the two coupled nonlinearoscillators considered in [261], showing how energy transfer is impeded by discreteness andnonlinearity. (a) Complete energy transfer when the first oscillator has initial amplitude a D 2:0,for E1 .0/ D 2:4) and (b) Incomplete energy transfer when the first oscillator has initial amplitudea D 2:5 and E1 .0/ � 4:1 (after [53])

Klein-Gordon (KG) system of ODEs written in the form

Run D �V 0 .un/C˛ .unC1 � 2un C un�1/ ; V .x/ D 1

2Kx2 C 1

4x4; �1 < n < 1

(7.1)where V.x/ is the on-site potential, ˛ > 0 is a parameter indicating the strength ofcoupling between nearest neighbors, and (0) denotes differentiation with respect tothe argument of V.x/.

It was not, however, until 1994, that a mathematical proof of the existence ofdiscrete breathers was published by MacKay and Aubry [235] in the case of one-dimensional lattices of the type (7.1). Under the general assumptions of nonlinearityand non-resonance, such chains of interacting oscillators were rigorously shownto possess discrete breather solutions for small enough values of the couplingparameter ˛, as a continuation of their obvious existence at ˛ D 0.

Section 7.2 below is devoted to a study of discrete breathers and multibreathersbased on the concept of homoclinic orbits of invertible maps. As has been pointedout in a number of papers [41, 42, 61, 62], homoclinic dynamics offers a veryconvenient way to construct such solutions and study their stability properties,away from the ˛ D 0 limit, through the “geometry” of the homoclinic solutions ofnonlinear recurrence relations. The results we shall describe have been presented in[39, 40], while for the most recent developments in this field the reader is stronglyadvised to consult the recent review article [125].

It is also important to remark at this point that, although strictly speaking discretebreathers are proved to exist in infinite lattices, in practice we construct them assolutions of a finite number of ODEs of the form (7.1), taking �N < n < N ,where N is large enough so that particles near the ends of the lattice are practicallymotionless.


−15

−0.5

−0.4

−0.3

−0.2

−0.1

un

(0)

−10 −5 0

n5 10 15

0

0.1

0.2

0.3

0.4

0.5

−15

−0.5

−0.4

−0.3

−0.2

−0.1

−10 −5 0

n5 10 15

0

0.1

0.2

0.3

0.4

0.5

−15

−0.4

−0.3

−0.2

−0.1un(0

)

un(0

)

−10 −5 0

n5 10 15

0

0.1

0.2

0.3

0.4

0.5

−15

−0.5

−0.4

−0.3

−0.2

−0.1

un

(0)

−10 −5 0

n5 10 15

0.1

0.2

0.3

0.4

0.5

0

−0.5

a b

dc

Fig. 7.2 Examples of discrete multibreathers of an FPU lattice of the form (6.1) with cubic ratherthan quadratic forces (the so-called FPU�ˇmodel), obtained by starting a Newton-Raphson searchusing homoclinic orbits of a map of the form (7.6) as an initial guess. The term ‘multibreather’refers to the fact that more than one site exhibit oscillations of comparably large amplitude (after[53])

7.2 Discrete Breathers and Homoclinic Dynamics

Focusing on the property of spatial localization, Flach was the first to show thatdiscrete breathers in simple one-dimensional chains can be accurately representedby homoclinic orbits in the Fourier amplitude space of time-periodic functions[123]. Indeed, inserting a Fourier series

un .t/ D1X

kD1An .k/ exp .ik!bt/ (7.2)

into the equations of motion of either the FPU or KG (7.1) lattice and setting theamplitudes of terms with the same frequency equal to zero, leads to the system ofequations

� k2!2bAn .k/ D h�V 0 .un/CW 0 .unC1 � un/�W 0 .un � un�1/ ; exp .Iik!bt/i;8k; n 2 Z; (7.3)

7.2 Discrete Breathers and Homoclinic Dynamics 169

where W is the potential function, I an infinite unit matrix and !b the fre-quency of the breather. This is an infinite-dimensional mapping of the Fouriercoefficients An .k/, with the brackets h:; :i indicating a normalized inner product.Time-periodicity is ensured by the Fourier basis functions exp .Iik!bt/. Spatiallocalization requires that An .k/ ! 0 exponentially as n ! ˙1. Hence a discretebreather is a homoclinic orbit in the space of Fourier coefficients, i.e. a doublyinfinite sequence of points beginning at 0 for n ! �1 and ending at 0 forn ! C1.

As a simple example, let us consider an infinite one-dimensional model whichplays an important role in many physical systems [188]: It is described by a specialform of the discrete nonlinear Schrodinger (DNLS) equation

iPun C � junj2un C �1C junj2

�.un�1 C unC1/ D 0; n 2 Z: (7.4)

Suppose now that we seek periodic solutions of the form un D xn exp.i!t/, withxn real. Substituting this expression of un in (7.4) and eliminating the exponentialexp.i!t/ leads to the two-dimensional map

xnC1 C xn�1 D ! � �x2n1C x2n

xn; xn 2 R: (7.5)

This map has three fixed points, in general, in the .xn; xnC1/ plane: One at the originand two located symmetrically along the diagonal xn D xnC1. The eigenvaluesof the linearized equations about these equilibria dictate e.g. that for ! D 0:25

and � D �3 the ones on the diagonal are saddles, while .0; 0/ is a center (seeExercise 7.1).

On the other hand, for ! D 4 and � D 1 the fixed point at the origin isa saddle, while the other two fixed points are centers. This is the case shownin Fig. 7.3, where the stable and unstable manifolds of .0; 0/ have been plotted.Clearly, their intersections form homoclinic orbits, which yield discrete breathers(and multibreathers) of the DNLS equation with frequency ! D !b as follows:Since, by definition, homoclinic orbits satisfy xn ! 0 as n ! ˙1, very few oftheir xn; xnC1 coordinates have appreciable magnitude, corresponding to sites unwhich are oscillating periodically with frequency !.

Now, if a homoclinic orbit is formed by intersections of the first part of the stable(resp. unstable) manifold with the first “lobes” of the unstable (resp. stable) manifoldof the saddle point at .0; 0/, this yields an orbit that makes one loop around theelliptic point and leads to oscillations with a single major amplitude belonging to asimple breather. If, on the other hand, the homoclinic orbit consists of intersectionsbetween “lobes” of both manifolds and makes several loops about an elliptic point,it leads to oscillations with large amplitudes at several sites and hence correspondsto a multibreather.

Of course, if the Fourier expansion of a breather consists of a single term, as inthe DNLS case, the associated two-dimensional map represents the exact dynamics.


−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

xn

y n

Fig. 7.3 The stable (dashed curve) and unstable (solid curve) manifolds of the fixed point at .0; 0/of a 2D DNLS map of the form (7.5), with yn D xnC1, or xn D an, yn D anC1 as in (7.6).The manifolds are clearly seen to intersect at infinitely many points, hence a wealth of homoclinicorbits exists (after [53])

What happens, however, in a case like the KG system (7.1), where the substitutionof (7.2) leads to an infinite dimensional map like (7.3) for the Fourier coefficients?Not to worry. Owing to the exponential decay of their magnitudes, we can get awaykeeping only the largest one (k D 1), reducing (7.3) to a simple 2D map, for an DAn.1/, of the form

anC1 D g .an; an�1/ : (7.6)

The invariant manifolds, emanating out of the saddle point at the origin, could thenbe plotted, as in Fig. 7.3 for example, by the method described in Exercise 7.1. It iseasy to apply this approach to the KG chain, where the above reduction yields themap

anC1 D �an�1 � Can C 1

˛a3n; C D �2C K � !2

˛(7.7)

(see Exercise 7.2 and Problem 7.1). As explained in [61], one can use this map tofind approximate analytical expressions for the coefficients an that will allow usto obtain numerically not only simple breathers but also multibreathers of the KGchain. Moreover, interesting effects arise when one applies this approach to KG“soft” potentials with a minus sign before the quartic term (see (7.1), [262]).


7.2.1 How to Construct Homoclinic Orbits

It is hopefully clear by now from the above discussion (as well as Exercises 7.1and 7.2), that the accurate computation of invariant manifolds and their intersectionsis extremely important for constructing discrete breathers and multibreathers. Forthis reason, we shall now describe a particularly useful and efficient method that hasbeen developed in [42] to locate homoclinic orbits of invertible maps of arbitrary(but finite) dimension. As it turns out, we need to exploit certain symmetry propertiesof the dynamics and understand the “geometry” of the invariant manifolds near theorigin, which we shall henceforth assume to be a saddle point of the maps understudy.

To see how this can be done, let us consider a general first order map (orrecurrence relation) of the form

xnC1 D f .xn/ ; xn; xnC1 2 Rd ; (7.8)

where d � 2 is a positive integer. Recall that homoclinic orbits are solutions whichsatisfy xn ! 0 as n ! ˙1 and concentrate on all orbits satisfying a symmetrycondition of the form

xn D Mx�n; (7.9)

where M is a d � d -matrix with constant entries and detM ¤ 0. Furthermore,observe that if an orbit obeys such a symmetry and xn ! 0 as n ! �1, it is alsotrue that xn ! 0 as n ! 1. Hence, any orbit which obeys this symmetry and alsosatisfies xn ! 0 as n ! �1 is a homoclinic orbit.

To construct such an orbit, we first need to specify numerically its asymptoticbehavior as n ! �1. In other words, it has to be on the unstable manifoldof x D 0. Thus, given that the origin is a saddle fixed point, it is known thatits unstable manifold is well approximated in its vicinity by the correspondingEuclidean subspace of the linearized equations, which is tangent to the nonlinearmanifold and has the same number of dimensions.

Now, the dimension of the linear unstable subspace equals the number ofcoordinates necessary to determine a point x�N , N � 1 uniquely. So, by choosingthis point to be on the linear unstable manifold, very close to the origin, it will alsolie approximately on the corresponding nonlinear manifold. Thus, when mappedforward N C 1 times, we can test whether it satisfies the above symmetry relation(7.9). By this reasoning, locating symmetric homoclinic orbits reduces to a searchfor solutions of the system

�x1 �M x�1 D 0

x0 �M x0 D 0; (7.10)

since, given x�N , the values of x1, x0 and x�1 are uniquely obtained by directiteration of the map (7.8).


Now, given that x1 D F.x0/ and x�1 D F �1.x0/, (7.10) can be solved for x0,searching for solutions of the equation

Z.x0/ D F.x0/�MF�1.x0/ D 0: (7.11)

The number of unknowns in this equation is 2d , d being the dimension of theunstable manifold. However, a zero of this function automatically implies x0 DM x0 (i.e. the second component of x0 is equal to zero, w0 D 0), reducing thenumber of unknowns to d .

Suppose now that xeq is a fixed point of the map (7.8) and a state x�N exists, withN � 1, for which x�N�n ! xeq as n ! 1, i.e. x�N is on the unstable manifold ofxeq . Thus, the point x�N can be uniquely identified by a d -dimensional coordinate,say � . Since x0 D FN .x�N / this also determines all the unknowns in (7.11), whichcan thus be written as an equation Z.� / D 0. Clearly, every coordinate � solvingthis equation defines an orbit of (7.8) having the desired asymptotic behavior. Thesearch for homoclinic and heteroclinic solutions, therefore, reduces to finding astate x�N on the unstable manifold of the equilibrium point xeq , determined bythe coordinate � , for which Z.� / D 0. This yields a set of d equations with dunknowns, and hence is, in general, solvable e.g. by Newton iterative methods (seeProblem 7.2 for more details).

In the case of an invertible map, we can use this method to also find allasymmetric homoclinic orbits, i.e. those which do not obey the symmetry condition(7.10). This can be done by introducing the new “sum” and “difference” variables

�vn D xn C x�nwn D xn � x�n

;

which always possess the symmetry

�vn D v�nwn D �w�n

: (7.12)

In this way, we can apply again the above strategy and look for symmetrichomoclinic orbits of a new map (whose dimension is twice that of the original f )described by the equations

F W(

vnC1 D f� vnCwn

2

� C f �1 � vn�wn2

�

wnC1 D f� vnCwn

2

� � f �1 � vn�wn2

� ; (7.13)

yielding homoclinic orbits xn of the original map f (7.8), that are not themselvesnecessarily symmetric. On the other hand, each homoclinic orbit of f is a symmetrichomoclinic orbit of the new map F . Therefore, by determining all symmetrichomoclinic orbits of F , we find all homoclinic orbits of f . In fact, it is possibleby this approach to classify all multibreathers of an N -particle one-dimensional


OACBOGEC

OC

-5-10-15

-3

-2

-1

0

0

n5 10 15

1

2

3

Xn

Xn

Xn

Xn

-5-10-15

-3

-2

-1

0

0

n5 10 15

1

2

3

-5-10-15

-3

-2

-1

0

0

n5 10 15

1

2

3

OGH

-5-10-15

-3

-2

-1

0

0

n5 10 15

1

2

3a b

c d

Fig. 7.4 Several homoclinic orbits of the map F (7.13), determined by a zero-search of the systemof equations v1 � v�1 D 0 and w0 D 0, related by vn D xn C x�n and wn D xn � x�n. Shownhere are vn (dashed), wn (dotted) and xn (solid). Also indicated is the symbolic name of the orbitassigned by the procedure outlined in the text and described in detail in [41, 42] (after [53])

lattice by assigning to them symbolic sequences in a systematic way, as follows:If, at a moment when all velocities are zero, a particle has a large positive (resp.negative) displacement, it is assigned a C (resp. �) sign, while if its displacement issmall it is assigned the symbol 0. We then introduce nine new symbols: A D �C,B D 0C, C D CC, D D �0, O D 00, E D C0, F D ��, G D 0� andH D C� corresponding to nine regions of the associated 2D map where thesepoints are located. One may thus locate multibreathers of increasing complexity,whose number, of course, grows with increasingN . The interested reader is referredfor more details to [41, 42] and Fig. 7.4 for an example of the results presented inthese papers.

It is important to mention that, besides the above approach based on homoclinicorbits, there also exist other methods to compute breathers and multibreathers thatone may find simpler or more straightforward to implement. Perhaps the mostpopular of them starts from the so called “anticontinuum limit”, where the particlesare uncoupled and uses Newton iterative schemes to continue the solutions to theregime of finite coupling [21, 248]. The homoclinic approach, however, also hasimportant advantages: It uses Fourier decomposition to turn the problem to aninvertible map whose low-dimensional approximations yield very accurate initial


conditions for the homoclinic orbits, provides a clear geometric and symbolicdescription of all breathers and multibeathers and can be used to locate suchsolutions in lattices of more than one dimension, as we demonstrate in the sectionthat follows.

7.3 A Method for Constructing Discrete Breathers

Based on the above results, we now describe a method for computing discretebreathers that can be efficiently applied to: (1) lattices of more than one spatialdimension and (2) systems with vector valued variables assigned to each lattice site.The main idea is to write a breather solution as the product of a space-dependentand a time-dependent part and reduce the problem to the computation of homoclinicorbits of a 2D map, under the constraint that the given ODEs possess simple periodicoscillations of a well-defined type and specified period.

As we have already discovered, it is possible to formulate a map in Fourieramplitude space linking discrete breathers to homoclinic orbits. This map canbe reduced to a finite-dimensional recurrence relation, by neglecting Fouriercomponents with a wave number k larger than some cutoff value kmax. Then, onecan use the methods described in [41,42] to approximate discrete breather solutionsby finding all homoclinic solutions of the corresponding recurrence relations.

This problem has also been examined from a different perspective: In 2002,inspired by the work of Flach [122] and Kivshar [189], Tsironis [334] suggesteda new way to approximately separate amplitude from time-dependence, yieldingin some cases ODEs with known solutions (for example elliptic functions) whilekeeping the dimension of the recurrence relation as low as possible. This led toan improved accuracy of the calculations and provided analytical expressions ofdiscrete breathers for a special class of FPU and KG systems.

In [39], Bergamin extended Tsironis’ work by developing a numerical procedurefor which the time-dependent functions need not be known analytically. In this way,a much wider class of nonlinear lattices can be treated involving scalar or vectorvalued variables in one or more spatial dimensions. In particular, the approximationproposed by Tsironis can be formulated as follows

�unC1 .t/ � un .t/ � .anC1 � an/ Tn .t/un�1 .t/ � un .t/ � .an�1 � an/ Tn .t/

; (7.14)

where an denotes the time-independent amplitude of un .t/ and Tn .t/ is its time-dependence, defined by Tn .0/ D 1 and PTn .0/ D 0.

Note now, that since all FPU and KG systems are described by a Hamiltonian ofthe form

H D 1

2

1X

nD�1Pu2n C

1X

nD�1fV .un/CW .unC1 � un/g ;

7.3 A Method for Constructing Discrete Breathers 175

we may use the approximation (7.14) to transform the equations of motion

Run D �V 0 .un/CW 0 .unC1 � un/ �W 0 .un � un�1/

into

an RTn D �V 0 .anTn/CW 0 ..anC1 � an/ Tn/�W 0 ..an � an�1/ Tn/ : (7.15)

This is an ODE for Tn .t/ which can, in principle, be solved since the initialconditions are known.

Of course, an analytical solution of (7.15) is in general very difficult to obtain.However, since we are primarily interested in the amplitudes an, what we ultimatelyneed to do is develop a numerical procedure to find a recurrence relation linking an,anC1 and an�1 without having to solve the ODE beforehand.

Let us observe first, that the knowledge of an, anC1 and an�1 permits us to solvethe above ODE numerically. Under mild conditions, solutions Tn .t/ can thus beobtained, which are time-periodic, while for a discrete breather all functions Tn .t/have the same period. Choosing a specific value for this period, allows us to invertthis process and determine anC1 as a function of an and an�1, just as we did in(7.6). In the same way, we also determine an�1 as a function of an and anC1. Thus,a two-dimensional invertible map has been constructed for the an, ensuring that alloscillators have the same frequency.

As is explicitly shown in [39,40], on a variety of examples, the homoclinic orbitsof this map provide highly accurate approximations to the discrete breather solutionswith the given period and the initial state un .0/ D an, Pun .0/ D 0. In fact, we cannow apply this approach to more complicated potentials and higher dimensionallattices, as demonstrated in Fig. 7.5, where we compute a discrete breather solutionof a two-dimensional lattice, with indices n, m in the x, y directions and dependentvariable un;m .t/.

Having thus discovered new and efficient ways of calculating discrete breathersin a wide class of nonlinear lattices, we may now turn to a study of their stability,continuation and control properties in parameter space. More specifically, we willshow that it is possible to use the above methods to extend the domain of existence ofbreathers to parameter ranges that cannot easily be accessed by other more standardcontinuation techniques.

7.3.1 Stabilizing Discrete Breathers by a Control Method

So far, we have seen that transforming nonlinear lattice equations to low-dimensional maps and using numerical techniques to compute their homoclinicorbits provides an efficient method for approximating discrete breathers in any(finite) dimension and classifying them in a systematic way. This clears the path foran investigation of important properties of large numbers of discrete breathers of


−10 −5 0 5 10010

−1

−0.5

0

0.5

1

1.5

mn

un

,m(0

)

Fig. 7.5 A breather in a two-dimensional lattice is obtained by the method described in the text,for equations of motion where the particles at each lattice site .n;m/ are subject to an on-sitepotential of the KG type and experience harmonic interactions with their four nearest neighbors(after [39])

increasing complexity. One such property, which is relevant to many applicationsand requires that a solution be known to great accuracy is (linear) stability in time.

In [62], the accurate knowledge of a discrete breather solution was used in arather uncommon way to study stability properties: As is well known, a familiartask in physics and engineering is to try to influence the behavior of a system, byapplying control methods. By control we refer here to the addition of an externalforce to the system which allows us to influence its dynamics. In particular, ourobjective is to use this force to change the stability type of a discrete breathersolution from what it was in the absence of control.

It is important to emphasize, of course, that during the application of control oursystem is altered in such a way that the solution itself does not change. In otherwords, the controlled system possesses solutions which are exactly the same as inthe uncontrolled case. Thus, if a solution is unstable in the uncontrolled system, theextra terms added to the equations by the external forcing can cause the solution tobecome stable and vice versa. This approach is often called feedback control andwas first used very successfully to influence the motion of dynamical systems byPyragas in [278].

The system that was studied in [62], using the above feedback control method, isa KG one-dimensional chain, whose equations of motion are written in the form

Run D �V 0 .un/C ˛ .unC1 � 2un C un�1/C Ld

dt.Oun � un/ ; (7.16)

where Oun D Oun.t/ is the known discrete breather solution of the equations whenL D 0. The parameter L indicates how strongly the control term influences the


−10 −8 −6 −4 −2 0 2 4 6 8 10

0

Position

Rel

ativ

e A

mp

litu

de

Fig. 7.6 A breather solution of (7.16), un .t/ D Oun .t/, which is unstable for L D 0 (after [62])

KG system. Thus, for any value of L, un.t/ D Oun.t/ is clearly a solution ofboth the controlled as well as the uncontrolled equations. Clearly, for L > 0,L d

dt un is a dissipative-like term, while L ddt

Oun represents a kind of periodic forcing.It is therefore reasonable to investigate whether, by increasing L, the dissipativepart of the process will force the system to converge to a stable solution. If thissolution is the Oun.t/ we started with, the latter is stable. If this is not the case, theoriginal solution is unstable. Indeed, it is not difficult to prove (see Exercise 7.3) thefollowing proposition:

Proposition 7.1. Let un D Oun.t/ be a periodic discrete breather solution of thelattice equations

Run D �V 0 .un/C ˛ .unC1 � 2un C un�1/ ; �N < n < N;

whereN is large enough so that the solution Oun.t/ exists. Then there exists anL > 0such that un D Oun.t/ is an asymptotically stable solution of the modified (controlled)system

Run D �V 0 .un/C ˛ .unC1 � 2un C un�1/C Ld

dt.Oun � un/ :

This result clearly implies that, by increasingL, it is possible to stabilize the originalsolution, independent of its stability character in the uncontrolled situation. Let usdemonstrate this by taking the breather of Fig. 7.6, which is unstable at L D 0,substitute its (known) solution form Oun .t/ in the above equations and increase thevalue of L. As we see in Fig. 7.7, it is quite easy to stabilize it at L D 1:17, since


−1 0 1 2 3 4 5−1

−0.5

0

0.5

1

real

imag

(a) L = 0.0

−1 0 1 2 3 4 5−1

−0.5

0

0.5

1

real

imag

(b) L = 1.10

−1 0 1 2 3 4 5−1

−0.5

0

0.5

1

real

imag

(c) L = 1.17

−1 0 1 2 3 4 5−1

−0.5

0

0.5

1

real

imag

(d) L = 1.25

Fig. 7.7 Increasing the control parameter L from L D 0 in the system (7.16), using as initialshape the one given in Fig. 7.6 moves the eigenvalues of the monodromy matrix of the orbit insidethe unit circle. Initially, some eigenvalues lie outside the unit circle, but eventually, for L > 1:17,all eigenvalues attain magnitudes less than one, thus achieving stability and control (after [62])

increasing the value of L gradually brings all eigenvalues of the monodromy matrixof the solution inside the unit circle.

The above Proposition has an additional significant advantage: It gives us theopportunity to address the question of the existence of discrete breathers in ranges ofthe coupling parameter ˛ where other more straightforward continuation techniquesdo not apply. In order to do this, it is important to recall how this question wasoriginally answered in the existence proof of MacKay and Aubry, using the notionof the so-called anti-continuum limit ˛ D 0 [235].

Let us observe that the KG equations of motion

Rxi D V 0 .x/C ˛ .xiC1 � 2xi C xi�1/ ;

describe a system of uncoupled oscillators for ˛ D 0. Obviously, in that case,any initial condition, where only a finite number of oscillators have a non-zeroamplitude, produces a discrete breather solution. In their celebrated paper of 1994,MacKay and Aubry prove that, under the conditions of nonresonance with thephonon band and nonlinearity of the function V 0 .x/, this solution can be continuedto the regime where ˛ > 0.


-1

-1

-0.8

-0.6

-0.4

-0.2

0

Imag

inar

y p

art

of

eig

enva

lues

0.2

0.4

0.8

1

0.6

-0.8 -0.6 -0.4 -0.2 0 0.2

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2

Real part of the eigenvalues

Imag

inar

y p

art

of

the

eig

enva

lues

0.4 0.6 0.8 1

0.4

Real part of eigenvalues

α=0.082, controlled system, L=0.13

α=0.08, uncontrolled system

0.6 0.8 1

-1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Imag

inar

y p

art

of

the

eig

enva

lues 0.6

0.8

1

-0.5 0 0.5

Real part of the eigenvalues

1 1.5

α=0.082, uncontrolled system

-1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Imag

inar

y p

art

of

eig

enva

lues

-0.8 -0.6-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Real part of eigenvalues

α=0.08, controlled system, L=0.13a b

c d

Fig. 7.8 (a) As the uncontrolled system approaches a bifurcation point at some critical value˛cr D 0:081 when L D 0, the bifurcation can be avoided as follows: (b) Fixing ˛ D 0:08

and increasing the control parameter, for example to L D 0:13, where the breather solution isasymptotically stable, (c) increasing the coupling strength to ˛ D 0:082 and (d) switching off thecontrol to return to the breather of the uncontrolled system, which is now unstable (after [62])

According to their approach, however, continuation for ˛ > 0 is possible, onlyas long as the eigenvalues of the Floquet matrix of the solution do not cross thevalue C1. This means that the typical occurrence of a bifurcation, through which thebreather becomes unstable, prevents such a continuation approach from followingthe breather beyond that critical ˛ value. This is where the control method proposedin [62] comes to the rescue: As we can see in Fig. 7.8, following a path in ˛ > 0

and L > 0 space, a discrete breather solution can be continued to a higher value of˛, by choosing L in the controlled system such that the solution remains stable!

Therefore, since by the above Proposition one can always find L such thatstability is possible for any coupling ˛, this allows the continuation of any solutionof the ˛ D 0 case to ˛ > 0 values beyond bifurcation, demonstrating the existenceof breather solutions in the corresponding parameter regime. The fact that Oun.t/—which is a stable solution of the controlled system—is by definition also an unstable


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Coupling parameter α

Co

ntr

ol p

aram

eter

L

Unstable region

Stable region

Fig. 7.9 Relation between the control parameter L and the coupling strength ˛: If ˛ is increased,L has to be larger to achieve stability and successful control. Shown here are the regions for whichthe breather of Fig. 7.6 is a stable or unstable solution un .t/ D Oun .t/ of (7.16) (after [62])

solution of the uncontrolled system, implies that we have succeeded in continuing adiscrete breather solution to higher values of ˛, beyond the bifurcation point.

In Fig. 7.9, we show in .L; ˛/ space the regions of stability of this particularbreather. As is well-known by the work of Segur and Kruskal [300], breathers arenot expected to exist in these systems in the continuum limit of ˛ going to infinity.At what coupling value though and how do they disappear? Can we use our controlaided continuation methods to follow them at arbitrarily high ˛ to answer suchquestions? These are open research problems that offer interesting applications ofthe methods we have described in this chapter.

7.4 Disordered Lattices

7.4.1 Anderson Localization in Disordered Linear Media

Let us now turn our attention to the topic of one-dimensional disordered lattices thathave applications to many physical problems.

One fundamental issue of interest to condensed matter physics was (and stillremains) the study of conductivity of electrons in solids. Since in an infiniteperfect crystal electrons can propagate ballistically, a natural question to ask is whathappens in a more realistic situation when disorder is present in the crystal due toimpurities or defects? Will an increase of the degree of disorder lead to a decreaseof conductivity, or not? These questions were first answered in a seminal paperby Philip Anderson [10], where it was shown that if disorder is large enough the

7.4 Disordered Lattices 181

diffusive motion of the electron will come to a halt. In particular, Anderson studiedan unperturbed lattice of uncoupled sites, where the perturbation was consideredto be the coupling between them, and randomness was introduced in the on-siteenergies. For this model he showed that for a large degree of randomness, thetransmission of a wave decays exponentially with the length of the lattice.

This absence of wave diffusion in disordered media is nowadays called Andersonlocalization, and is a general wave phenomenon that applies to the transportof different types of classical or quantum waves, like electromagnetic, acousticand spin waves. Its origin is the wave interference between multiple scatteringpaths; i.e. the introduction of randomness can drastically disturb the constructiveinterference, leading to the halting of waves. Anderson localization plays animportant role in several physical phenomena. For example, the localization ofelectrons has dramatic consequences regarding the conductivity of materials, sincethe medium no longer behaves like a metal, but becomes an insulator when thestrength of disorder exceeds a certain threshold. This transition is often referred asthe metal-insulator transition.

Although today the significance of Anderson localization is well recognized asindicated for example by the large number of theoretical and numerical papersrelated to this topic, its physical relevance was not fully realized for many years.The fact that experimental observation of Anderson localization was (and still is) acumbersome task played an important role in this situation. It is worth mentioningthat Anderson himself failed initially to understand the significance of his discovery,as he admits in his Nobel price winning lecture in 1978, where he stated aboutlocalization that ‘. . . very few believed it at the time, and even fewer saw itsimportance, among those who failed to fully understand it at first was certainlyits author. It has yet to receive adequate mathematical treatment, and one has toresort to the indignity of numerical simulations to settle even the simplest questionsabout it.’

Many theoretical and numerical approaches of localization start with theAnderson model: a standard tight-binding (i.e. nearest-neighbor hopping) systemwith on-site potential disorder. This can be represented in one dimension by atime-dependent Schrodinger equation

i@ l

@tD �l l � lC1 � l�1; (7.17)

where �l are the random on-site energies, drawn from an uncorrelated uniformdistribution in Œ�W=2;W=2�, where W parameterizes the disorder strength and lis the complex wave function associated with lattice site l . Using the substitution l D Al exp.�i�t/ yields a time-independent system of difference equations

�Al D �lAl � AlC1 �Al�1; (7.18)

whose solution consists of a set of eigenvectors called the normal mode eigen-vectors (NMEs), A�l , (normalized as

Pl .A

�l /2 D 1 ), and a set of eigenvalues

called the normal frequencies, �� . All eigenvectors are exponentially localized,


meaning that their asymptotic behavior can be described by an exponential decayjA�l j � e�l=�.��/, where �.��/ is a characteristic energy-dependent length, calledthe localization length. Naturally, � ! 1 corresponds to an extended eigenstate.Several approaches have been developed for the evaluation of �, such as the transfermatrix method, schemes based on the transport properties of the lattice, as well asperturbative techniques. For more information on such approaches, the reader isreferred to [206] and references therein.

The mathematical description of Anderson localization based on the aboveformalism is useful for theoretical approaches, but not so much for experimentalstudies. Clearly, unbounded media do not exist and eigenvalues or eigenstates arerarely measured in real experiments, where mainly measurements of transition andconductivity are performed. So the need for a connection between conductivityand the spectrum of the system becomes apparent. The basic approach towards thefulfilment of this goal was the establishment of a relationship between conductivityand the sensitivity of the eigenvalues of the Hamiltonian of a finite (but very large)system to changes in the boundary conditions [118]. This sensitivity turned out tobe conceptually important for the formulation of a scaling theory for localization[5]. The main hypothesis of this single-parameter scaling theory is that, close tothe transition between localized and extended states, there should only be onescaling variable which depends on the conductivity for the metallic behavior andthe localization length for the insulating behavior. This single parameter turned outto be a dimensionless conductance (often called Thouless conductance or Thoulessnumber) defined as

g.N / D ıE

�E; (7.19)

where ıE is the average energy shift of eigenvalues of a finite system of size Ndue to the change in the boundary conditions, and �E is the average spacing ofthe eigenvalues. For localized states and largeN , ıE becomes very small and g.N /vanishes, while in the metallic regime the boundary conditions always influence theenergy levels, even in the limiting case of infinite systems. The introduction of theThouless conductance led to the formulation of a simple criterion for the occurrenceof Anderson localization:g.N / < 1. In one and two-dimensional random media thiscriterion can be reached for any degree of disorder by just increasing the size of themedium, while in higher dimensions a critical threshold exists.

The experimental verification of Anderson localization is not easy, for exampledue to the electron–electron interactions in cases of electron localization, andthe difficult discrimination between localization and absorption in experiments ofphoton localization. Nevertheless, nowadays the observation of Anderson local-ization has been reported in several experiments, a few of which we quote here:(1) light localization in three-dimensional random media [323, 345], (2) transverselocalization of light for two [299] and one [208] dimensional photonic lattices, (3)localization of a Bose-Einstein condensate in an one-dimensional optical potential[45, 284], and (4) elastic waves propagating in a three-dimensional disorderedmedium [173]. In addition, the observation of the metal-insulator transition in athree-dimensional model with atomic matter waves has been described in [78].


7.4.2 Diffusion in Disordered Nonlinear Chains

As was discussed in the previous subsection, the presence of uncorrelated spatialdisorder in one-dimensional linear lattices results in the localization of theirNMEs. An interesting question, therefore, that naturally arises is what happensif nonlinearity is introduced to the system. Understanding the effect of nonlin-earity on the localization properties of wave packets in disordered systems is achallenging task, which, has attracted the attention of many researchers in recentyears [47, 48, 124, 131, 144, 202, 210, 252–255, 271, 311, 317, 338, 339]. Most ofthese studies consider the evolution of an initially localized wave packet and showthat wave packets spread subdiffusively for moderate nonlinearities. On the otherhand, for weak enough nonlinearities, wave packets appear to be frozen over thecomplete available integration time, thereby resembling Anderson localization, atleast on finite time scales. Recently, it was conjectured in [178] that these statesmay be localized for infinite times on Kolmogorov-Arnold-Moser (KAM) torus-likestructures in phase space. Whether this is true or not remains open to the present day.

7.4.2.1 Two Basic Models

In order to present the characteristics of subdiffusive spreading, let us considertwo one-dimensional lattice models. The first one is a variant of system (7.4). Itrepresents a disordered discrete nonlinear Schrodinger equation (DDNLS) describedby the Hamiltonian function

HD DX

l

�l j l j2 C ˇ

2j l j4 � . lC1 �

l C �lC1 l /; (7.20)

in which l are complex variables, �l their complex conjugates, l are the lattice

site indices and ˇ � 0 is the nonlinearity strength. The random on-site energies�l are chosen uniformly from the interval

��W2; W2

�, with W denoting the disorder

strength. The equations of motion are generated by P l D @[email protected] ?l /:

i P l D �l l C ˇj l j2 l � lC1 � l�1: (7.21)

This set of equations conserves both the energy of (7.20), and the norm S

D Pl j l j2.

The second model we consider is the quartic KG lattice (7.1), described by theHamiltonian

HK DX

l

p2l2

C Q�l2

u2l C 1

4u4l C 1

2W.ulC1 � ul /

2; (7.22)


where ul and pl respectively are the generalized coordinates and momenta on site l ,and Q�l are chosen uniformly from the interval

�12; 32

�. The equations of motion are

Rul D �@HK=@ul and yield

Rul D �Q�lul � u3l C 1

W.ulC1 C ul�1 � 2ul /: (7.23)

This set of equations only conserves the energyH of (7.22). The scalar valueH � 0

serves as a control parameter of nonlinearity, similar to ˇ for the DDNLS case.For ˇ D 0 and l D Al exp.�i�t/, (7.21) reduces to the linear eigenvalue

problem (7.18). The width of the eigenfrequency spectrum �� in (7.18) is �D DW C 4 with �� 2 ��2 � W

2; 2C W

2

�. In the limit H ! 0 (in practice by neglecting

the nonlinear term u4l =4) the KG model of (7.22) is reduced (with ul D Al exp.i!t/)to the same linear eigenvalue problem of (7.18), under the substitutions � D W!2�W � 2 and �l D W.Q�l � 1/. The width of the squared frequency !2� spectrum is�K D 1C 4

Wwith !2� 2 �

12; 32

C 4W

�. As in the case of DDNLS, W determines the

disorder strength.In the case of weak disorder, W ! 0, the localization length of NMEs is

approximated by �.��/ � �.0/ � 100=W 2 [206, 207]. On average the NMElocalization volume (i.e. spatial extent) V , is of the order of 3:3�.0/ � 330=W 2

for weak disorder and unity in the limit of strong disorder, W ! 1 [207]. Theaverage spacing of eigenvalues of NMEs within the range of a localization volumeis given by d � �=V , with � being the spectrum width. The two frequency scalesd � � determine the packet evolution details in the presence of nonlinearity.

In order to write the equations of motion of Hamiltonian (7.20) in the normalmode space of the system, we insert l D P

� A�;l� in (7.21), with j� j2denoting the time-dependent amplitude of the �th mode. Then, using (7.18) andthe orthogonality of NMEs the equations of motion (7.21) read

i P� D �� C ˇX

�1;�2;�3

I�;�1;�2;�3��1�2�3 (7.24)

with I�;�1;�2;�3 D Pl A�;lA�1;lA�2;lA�3;l being the so-called overlap integral (see

Exercise 7.4).To study the spreading characteristics of wave packets let us order the NMEs

by increasing value of the center-of-norm coordinate X� D Pl lA

2�;l [131, 210,

311, 317]. Then, for DDNLS we follow the normalized norm density distributionsz� j�j2=P

jj2, while for KG we monitor the normalized energy density

distributions z� E�=P

E with E� D PA2�=2 C !2�A2�=2, where A� is the

amplitude of the �th normal mode and !2� its squared frequency. Usually thesedistributions are characterized by means of the second momentm2 D P

�.�� N�/2z�(which quantifies the wave packet’s spreading width), with N� D P

� �z� , and theparticipation number P D 1=

P� z2� , (i.e. the number of the strongest excited

modes in z�). Another often used quantity is the compactness index � D P2=m2,which quantifies the inhomogeneity of a wave packet. Thermalized distributions


have � � 3, while � 3 indicates very inhomogeneous packets, e.g. sparse (withmany holes) or partially selftrapped ones (see [317] for more details).

The frequency shift ı of a single site oscillator induced by the nonlinearity forthe DDNLS model is ıD D ˇj l j2, while for the KG system the squared frequencyshift of a single site oscillator is ıK � El , El being the energy of the oscillator.Since all NMEs are exponentially localized in space, each effectively couples to afinite number of neighbor modes. The nonlinear interactions are thus of finite range;however, the strength of this coupling is proportional to the norm (energy) densityfor the DDNLS (KG) model. If the packet spreads far enough, we can generallydefine two norm (energy) densities: one in real space, nl D j l j2 (El) and the otherin normal mode space, n� D j� j2 (E�).

Typically, in order to obtain a statistical description of the wave packet dynamicsaveraging over hundreds of realizations is performed. As a result, no strongdifference is seen between the norms (energies) in real and normal mode space, andtherefore, we treat them generally as some characteristic norm (n) or energy (E)density. The frequency shift due to nonlinearity is then ıD � ˇn for the DDNLSmodel, while the square frequency shift is ıK � E for the KG lattice.

7.4.2.2 Regimes of Wave Packet Spreading

Let us now discuss in more detail the different wave packet evolutions. For strongnonlinearities a substantial part of the wave packet is self-trapped. This is dueto nonlinear frequency shifts, which will tune the excited sites immediately outof resonance with the non-excited neighborhood [125, 127]. The existence of theselftrapping regime was theoretically predicted for the DDNLS model in [202] (seealso [317] for more details). According to the theorem stated in [202], for largeenough nonlinearities (ıD > �D), single site excitations cannot uniformly spreadover the entire lattice. Consequently, a part of the wave packet will remain localized,although the theorem does not prove that the location of this inhomogeneity isconstant in time. In fact, partial self-trapping will occur already for ıD � 2

(ıK & 1=W ) since at least some sites in the packet may be tuned out of resonance.The selftrapping regime has been numerically observed for single-site excitations[131, 311, 317] and for extended excitations [47, 210], both for the DDNLS and theKG models, despite the fact that the KG system conserves only the total energyH ,and the selftrapping theorem can not be applied there.

When selftrapping is avoided for ıD < 2 (ıK . 1=W ), two different spreadingregimes were predicted in [124] having different dynamical characteristics: anasymptotic weak chaos regime, and a potential intermediate strong chaos one. Thebasic assumption for this prediction is that the spreading of the wave packet is dueto the chaoticity inside the packet. This practically means that a normal mode in alayer of width V in the cold exterior of the wave packet—which borders the packetbut will belong to the core of the spreading packet at later times—is incoherentlyheated by the packet. Numerical verifications of the existence of these regimes werepresented in [47, 210, 311].


Let us take a closer look at these behaviors by considering initial “block” wavepackets, where L central oscillators of the lattice are excited having the same norm(energy). In the weak chaos regime, for L � V and ı < d most of the NMEsare weakly interacting with each other. Then the subdiffusive spreading of the wavepacket is characterized by m2 � t1=3. If the nonlinearity is weak enough to avoidselftrapping, yet strong enough to ensure ı > d , the strong chaos regime is realized.Wave packets in this regime initially spread faster than in the case of weak chaos,withm2 � t1=2. Note that a spreading wave packet launched in the regime of strongchaos will increase in size, drop its norm (energy) density, and therefore a crossoverinto the asymptotic regime of weak chaos will occur at later times.

We turn our attention now to the case L < V . In this case the wave packetinitially spreads over the localization volume V during a time interval �in � 2 =d ,even in the absence of nonlinearities [124, 210]. The initial average norm (energy)density nin (Ein) of the wave packet is then lowered to n.�in/ � ninL=V (E.�in/ �EinL=V ). Further spreading of the wave packet in the presence of nonlinearitiesis then determined by these reduced densities. Note that for single-site excitations(L D 1), the strong chaos regime completely disappears and the wave packetevolves either in the weak chaos or selftrapping regimes [131, 271, 317, 338].

7.4.2.3 Numerical Results

We present now some numerical results obtained in [210] showing the existence andthe main characteristics of weak chaos, strong chaos, the crossover between themand the selftrapping regime (Fig. 7.10). These results were obtained by consideringcompact DDNLS wave packets at t D 0 spanning a width L centered in thelattice, such that within L there is a constant norm density and a random phaseat each site (outside the volume L the norm density is zero). In the KG case, thiscorresponds to exciting each site in the width L with the same energy density,E D H=L, i.e. setting initial momenta to pl D ˙p

2E with randomly assignedsigns. Ensemble averages over disorder were calculated for 1; 000 realizations withW D 4, while L D V D 21, and system sizes of 1; 000 � 2; 000 sites wereconsidered. For the DDNLS, an initial norm density of nin D 1 was taken, sothat ıD D ˇ. The values of ˇ (E for KG) were chosen to give the three expectedspreading regimes, respectively ˇ 2 f0:04; 0:72; 3:6g and E 2 f0:01; 0:2; 0:75g.

Let us describe the results of Fig. 7.10 referring mainly to the DDNLS model. Inthe regime of weak chaos we see a subdiffusive growth ofm2 according to m2 � t˛

with ˛ � 1=3 at large times. In the regime of strong chaos we observe exponents˛ � 1=2 for 103 . t . 104 (KG: 104 . t . 105). Time averages in these regionsover the green curves yield ˛ � 0:49 ˙ 0:01 .KG: 0:51 ˙ 0:02/. With spreadingcontinuing in the strong chaos regime, the norm density in the packet decreases, andeventually satisfies ıD � dD .ıK � dK ). This results in a dynamical crossover to theslower weak chaos subdiffusive spreading. Fits of this decay suggest that ˛ � 1=3

at 1010 . t . 1011 for both models. In the regimes of weak and strong chaos, the


2

3

4

5

⟨ log

10 m

2⟩

6 42log10 t log10 t

02 4

0.2

0.4

0.6

α

6 8

log10 t log10 t

02 4 6 2 4 6

1

2

3

4

⟨ζ⟩

80

1

2

3

4

⟨ζ⟩

Fig. 7.10 Upper row: Average log of second moments (inset: average compactness index) vs. logtime for the DDNLS (KG) on the left (right), for W D 4, L D 21. Colors correspond to thethree different regimes: (1) weak chaos—blue, ˇ D 0:04 .E D 0:01/, (2) strong chaos—green,ˇ D 0:72 .E D 0:2/, (3) self-trapping—red, ˇ D 3:6 .E D 0:75/. The respective lightersurrounding areas show one-standard-deviation error. Dashed lines are to guide the eye to � t 1=3,while dot-dashed lines are guides for � t 1=2. Lower row: Finite difference derivatives ˛.log t / Dd hlogm2i =d log t for the smoothed m2 data respectively from above curves (after [210])

compactness index at largest computational times is � � 2:85˙ 0:79 .KG: 2:74˙0:83/, as seen in the blue and green curves of the insets of Fig. 7.10. This means thatthe wave packet spreads, but remains rather compact and thermalized (� � 3).

Numerical studies of several additional models of disordered nonlinear one-dimensional lattices demonstrate that in the presence of nonlinearities subdiffusivespreading is always observed, so that the second moment grows initially asm2 � t˛

with ˛ < 1, showing signs of a crossover to the asymptotic m2 � t1=3 law atlarger times [47, 210]. Remarkably, subdiffusive spreading was also observed forlarge disorder strengths, when the localization volume (which defines the numberof interacting partner modes) tends to one [47]. Such results support the conjecturethat the wave packets, once they spread, do so up to infinite times in a subdiffusiveway, bypassing Anderson localization of the linear wave equations. Nevertheless,the validity of this conjecture is still an open issue.

It is worth-mentioning that when the nonlinearity strength tends to very small val-ues, waiting times for wave packet spreading of compact initial excitations increasebeyond the detection capabilities of current computational tools. The corresponding


question of whether a KAM regime can indeed be approached at finite nonlinearitystrength was addressed in [178], but remains still unanswered.

Exercises

Exercise 7.1. (a) Identify all fixed points of the two-dimensional map (7.5) asfunctions of the parameters ! and � . Linearize the equations of the map aboutthem and compute the eigenvalues and eigenvectors of the linearized map inevery case, thus determining the stability of each point.

(b) Choose ! and � values such that the origin is a saddle, place many initialconditions on the corresponding eigenvectors and trace out numerically thestable and unstable manifolds emanating out of the point .0; 0/ in that case(note that to follow the stable manifold you will need to use the equations of theinverse map).

(c) Computing as accurately as possible one of their primary intersections locateone homoclinic point, whose (forward and backward) iterations should enableyou to compute one of the simple breathers of the DNLS equation.

Exercise 7.2. (a) Substituting (7.2) in the KG equation of motion (7.1) derivethe infinite dimensional map satisfied by the Fourier coefficients An.k/ for aperiodic solution of the system with frequency !. Using these equations provethat the origin is of the saddle type provided the following condition is satisfied

C.k/ D 2C K � !2k2

˛> 2; (7.25)

thus showing that breathers can exist in the KG system with frequency !b D !

if !2k2 >KC4˛ for all kD 1; 2; : : :. It follows, therefore, that if this conditionis satisfied for k D 1 it will also hold for all other k.

(b) Referring to (7.7), with C D C.1/, consult reference [61] and use the methodproposed in Exercise 7.1 to compute homoclinic orbits for the cubic map (7.7)that approximate simple breathers of the KG system. Consult also Problem 7.1below to see how to do this in a systematic way.

Exercise 7.3. Prove Proposition 7.1 of Sect. 7.3.1 reasoning as follows: Let x DOx.t/ denote a periodic solution of

Rx D f .x/: (7.26)

We wish to show that there exists an Lc such that for L > Lc , x D Ox.t/ is anasymptotically stable solution of the system

Rx D f .x/C Ld

dt.Ox � x/: (7.27)


To this end, linearize (7.26), writing x D Ox.t/ C ".t/ and perform the change ofvariables ".t/ D �.t/ exp.�Lt

2/, transforming the linear system into a Hill’s type

equation for �.t/, involving the Jacobian matrix Df.Ox.t//, whose coefficients areperiodic in time. Now use the fact that the solution of Hill’s equation can be writtenas a linear combination of (n-dimensional) vector periodic functions Pk.t/; k D1; 2; : : : ; n multiplied by exponentials in the Floquet exponents, ˇk . Examining thenature of these exponents, show that the solution x D Ox.t/ is stable if the controlparameter L > 2jˇmaxj, where ˇmax is the largest in absolute value of all real ˇk ,since then the function ".t/ will tend to zero as time goes to infinity. Hint: Consult[240, 250, 344].

Exercise 7.4. Substituting l D P� A�;l� in (7.21), and using (7.18) and the

orthogonality of NMEs, prove that the equations of motion of the DDNLS system(7.20) in normal mode space is given by (7.24).

Problems

Problem 7.1. (a) Consider the “soft spring” KG potential

V.x/ D 1

2Kx2 � 1

4x4: (7.28)

Use this form of V.x/ in the equations of motion (7.1) to search for periodicsolutions of the form (7.2) with !b D !, equating coefficients of exp.ik!t/.Thus, derive the following algebraic system for the coefficients An.k/

AnC1.k/CAn�1.k/ D C.k/An.k/� 1

˛

X

k1

X

k2

X

k3

An.k1/An.k2/An.k3/;

k1 C k2 C k3 D k; (7.29)

for every k D 1; 2; : : : ;M , with

C.k/ D 2C K � k2!2

˛(7.30)

assuming that the Fourier coefficients An.k/ decrease sufficiently rapidly withn, so that M of them are enough to describe the dynamics. As explained inSect. 7.2, a necessary condition for the existence of breathers is that the trivialsolution of (7.29), i.e. An.k/ D 0, for all n; k, be a saddle point of the 2M -dimensional map (7.29).

(b) Linearize (7.29) near the trivial point and prove that it is a saddle with anM -dimensional stable and an M -dimensional unstable manifold, if and onlyif jC.k/j > 2 for all k D 1; 2; : : : ;M . Show that the smallest value of


! that makes it possible for a breather with this frequency to exist is ! DpK � ˛.C.1/� 2/. Consider now the simple approximation that the breather

is represented by a single mode only, i.e. un.t/ D 2An.1/cos!t , for �M <

n < M , and scale the Fourier coefficients byAn.1/ D p˛An to obtain the map

AnC1 C An�1 C C.1/An D �3A3n; (7.31)

(see also (7.7)).(c) Analyzing (7.31), show that the only possibility for breathers is C.1/ > 2,

with frequency ! <pK below the phonon band. Explain why the parameter

range for breathers of (7.28) is considerably more limited than the “hardspring” case (see (7.1)), where ! >

pK C 4˛. Write the above map in

the two-variable form AnC1 D �Bn � C.1/An � 3A3n; BnC1 D An anddetermine the eigenvectors of its linearized equations about the origin .0; 0/.Now use the method described in Exercise 7.1 to trace out numerically theinvariant manifolds of the origin, locate their intersections and compute someof the breathers and multibreathers. Start with accurate representations of theirun.0/ D 2An.1/ and Pun.0/ D 0 initial conditions and integrate the equations ofmotion (see (7.1)) for long times to determine numerically the stability of thesesolutions, under small changes of their initial conditions.

Problem 7.2. (a) Use the approach outlined in Sect. 7.2.1 to compute homoclinicorbits of the map (7.5), corresponding to symmetric breathers and multi-breathers of the DNLS (7.4), as follows: Solve (7.10), noting that, once x�N isknown as an initial condition, the values of x1, x0 and x�1 are uniquely obtainedby direct iteration of the map (7.8). Observe next that setting x1 D F.x0/ andx�1 D F�1.x0/, (7.10) can be solved for x0, by looking for solutions of the(7.11), whose number of unknowns is d , since Z.x0/ D 0 implies x0 D M x0(i.e. the second component of x0 is zero, w0 D 0).

(b) Express now the desired solution x�N , lying on the d -dimensional unstablemanifold, by a coordinate � satisfying Z.� / D 0. Since x0 D FN .x�N / thisalso completely determines all the unknowns in (7.11) and hence every solutionof this equation defines an orbit of (7.8) having the desired asymptotic behavior.One convenient way to define the unknown � D .�1; : : : ; �d / is to approximatethe state x�N by a point lying on the Euclidean unstable manifold of thelinearized equations and express it as a linear combination of the eigenvectorsEui , writing x�N D "

PdiD1 �iEu

i C xeq , where j�i j > 1 are the correspondingeigenvalues and j�j 1 a scalar parameter denoting the accuracy of thecomputation. Thus, you may now substitute this expression in Z.� / D 0 andsolve for the resulting d equations with d unknowns, using an appropriateNewton iterative method.

Chapter 8The Statistical Mechanics of Quasi-stationaryStates

Abstract This chapter adopts an altogether different approach to the study ofchaos in Hamiltonian systems. We consider, in particular, probability distributionfunctions (pdfs) of sums of chaotic orbit variables in different regions of phasespace, aiming to reveal the statistical properties of the motion in these regions. Ifthe orbits are strongly chaotic, these pdfs tend to a Gaussian and the system quicklyreaches an equilibrium state described by Boltzmann-Gibbs statistical mechanics.There exist, however, many interesting regimes of weak chaos characterized bylong-lived quasi-stationary states (QSS), whose pdfs are well-approximated byq-Gaussian functions, associated with nonextensive statistical mechanics. In thischapter, we study such QSS for a number of N dof FPU models, as well as 2D area-preserving maps, to locate weakly chaotic QSS, investigate the complexity of theirdynamics and discover their implications regarding the occurrence of dynamicalphase transitions and the approach to thermodynamic equilibrium, where energy isequally shared by all degrees of freedom of the system.

8.1 From Deterministic Dynamics to Statistical Mechanics

As the reader has undoubtedly noticed, we have regarded so far in this book thesolutions of Hamiltonian systems as individual trajectories (or orbits), evolving ina deterministic way, according to Newton’s equations of motion. In other words,we have systematically chosen initial conditions locally in various areas of phasespace and have sought to characterize the resulting orbits as “ordered” or “chaotic”,aiming to understand the dynamics more globally in phase space. By order, we meanthat the corresponding domain is predominantly occupied by invariant tori, whilechaos implies that extremely sensitive dependence on initial conditions prevails inthe region under study.

Clearly, however, if we seriously wish to shift our attention from local to globaldynamics, such characterizations are too simplistic. As we have already seen in


191

192 8 The Statistical Mechanics of Quasi-stationary States

earlier chapters, order and chaos are globally much more complex than what theirlocal manifestations might entail. In Chap. 6, for example, we realized that s-dimensional invariant tori of N dof Hamiltonians, with s � N , owing to theirlocalization properties in Fourier space, turn out to be very important in our attemptto understand the (global) phenomenon of FPU recurrences. Indeed, quasiperiodicorbits of as few as s D 2; 3; : : : (rationally independent) frequencies were foundto dominate the dynamics at low energies to the extent that, even when theseorbits become unstable and the corresponding tori break down, they still representquasi-stationary states, which preclude the onset of energy equipartition for verylong times.

A different kind of localized solutions (this time in configuration space), knownas discrete breathers, was discussed in Chap. 7. They were also found to hinderenergy equipartition, through the persistence of stable periodic oscillations, inwhich very few particles participate with appreciable amplitudes! Indeed, discretebreathers are observed in many cases to be surrounded by low-dimensional tori andclearly constitute one more example where local order influences global dynamicsin multi-dimensional Hamiltonian systems.

One may, therefore, well wonder: since there do exist hierarchical levels of order,could there also exist hierarchies of disorder, where chaotic orbits are confined,for very long times, in limited domains of phase space? Such regimes may becharacterized, for example, by small LCEs and weakly chaotic dynamics, comparedwith regimes of strong chaos, which are larger and possess higher LCEs. As we willshow in the present chapter, such distinction can indeed be made, but we must beprepared to combine our deterministic methods with the probabilistic approach ofstatistical mechanics.

As is well-known, the statistical analysis of dynamical systems has a long history.Probability density functions (pdfs) of chaotic orbits have been studied for manydecades and by many authors, aiming to understand the transition from deterministicto stochastic (or ergodic) behavior [11, 20, 117, 184, 185, 266, 268, 289–291, 308].In this context, the fundamental question that arises concerns the existence of anappropriate invariant probability density (or ergodic measure), characterizing phasespace regions where solutions generically exhibit chaotic dynamics. If such aninvariant measure can be established for almost all initial conditions (i.e. exceptfor a set of measure zero), one has a firm basis for studying the system at hand froma statistical mechanics point of view.

If, additionally, this invariant measure turns out to be a continuous and suf-ficiently smooth function of the phase space coordinates, one can invoke theBoltzmann-Gibbs (BG) microcanonical ensemble and attempt to evaluate all rel-evant quantities at thermal equilibrium, like partition function, free energy, entropy,etc. On the other hand, if the measure is absolutely continuous (as e.g. in thecase of the so-called Axiom A dynamical systems), one might still be able to usethe formalism of ergodic theory and Sinai-Ruelle-Bowen measures to study thestatistical properties of the model [117].

In all these cases, viewing the values of one, or a linear combination of compo-nents of a chaotic trajectory at discrete times tn, n D 1; : : : ; N as realizations of

8.1 From Deterministic Dynamics to Statistical Mechanics 193

several independent and identically distributed random variables Xn and calculatingthe distribution of their sums in the context of the Central Limit Theorem (CLT)[282] one expects to find a Gaussian, whose mean and variance are those of the Xn’s.This is indeed what happens in many chaotic dynamical systems studied to datewhich are ergodic, i.e. almost all their orbits (except for a set of measure zero) passarbitrarily close to any point of the constant energy manifold, after sufficiently longtimes. What is also true is that, in these cases, at least one LCE is positive, stableperiodic orbits are absent and the constant energy manifold is covered uniformly bychaotic orbits, for all but a (Lebesgue) measure zero set of initial conditions.

But then, what about chaotic regions of limited extent, at energies where theMLE is “small” and stable periodic orbits are present, whose islands of invarianttori and sets of cantori around them occupy a positive measure subset of the energymanifold? In such regimes of weak chaos, it is known that many orbits “stick” forlong times to the boundaries of these islands and chaotic trajectories diffuse slowlythrough multiply connected regions in a highly non-uniform way [8, 89, 249]. Suchexamples occur in many physically realistic systems studied in the current literature(see e.g. [131, 177, 316, 317]).

These are cases where exponential separation of nearby solutions is not uniform,exponential decay of correlations is not generic and chaotic orbits can no longer beviewed as independent and/or identically distributed random variables. What kindof pdfs would we expect from a computation of their sums, if not Gaussians? Thistype of Hamiltonian dynamics is strongly reminiscent of many examples of physicalsystems governed by long range interactions, like self-gravitating systems of finitelymany mass points, interacting black holes and ferromagnetic spin models, in whichpower laws are dominant over exponential decay [332].

8.1.1 Nonextensive Statistical Mechanics and q-Gaussian pdfs

As we also discussed in Sect. 1.4, multi-particle systems belong to differentuniversality classes, according to their statistical properties at equilibrium. In themost widely studied class, if the system can be at any one of i D 1; 2; : : : ; W stateswith probability pi , its entropy is given by the BG formula

SBG D �k

WX

iD1

pi ln pi ; (8.1)

where k is Boltzmann’s constant, under the constraint

WX

iD1

pi D 1: (8.2)


As is well-known, the BG entropy is additive in the sense that, for any twoindependent systems A and B , the entropy of their sum is the sum of the individualentropies, i.e. SBG.A C B/ D SBG.A/ C SBG.B/. It is also extensive, as it growslinearly with the number of dof N , as N ! 1. These properties are associatedwith the fact that different parts of BG systems are highly uncorrelated and theirdynamics is statistically uniform in phase space.

There is, however, an abundance of physical systems characterized by strongcorrelations, for which the assumptions of extensivity and additivity are notgenerally valid [332]. In fact, as we explain in this chapter, Hamiltonian systemsprovide a wealth of examples governed by such complex statistics, especially nearthe boundaries of islands of ordered motion, where orbits “stick” for very long timesand the dynamics becomes very weakly chaotic. It is for this kind of situations thatTsallis proposed the entropy formula

Sq D k1 �PW

iD1 pqi

q � 1with

WX

iD1

pi D 1; (8.3)

that depends on an index q, for a set of W states with probabilities pi i D 1; : : : ; W ,obeying the constraint (8.2). The Sq entropy is not additive, since Sq.A C B/ DSq.A/CSq.B/Ck.1�q/Sq.A/Sq.B/ and generally not extensive. The pdf replacingthe Gaussian in this type of nonextensive statistical mechanics is the q-Gaussiandistribution

P.s/ D a expq.�ˇs2/ � a

�1 � .1 � q/ˇs2

� 11�q

(8.4)

obtained as an extremum (maximum for q > 0 and minimum for q < 0) of theTsallis entropy (8.3), under appropriate constraints [332]. The q index satisfies 1 <

q < 3 to make (8.4) normalizable, ˇ is an arbitrary parameter and a a normalizationconstant. Note that in the limit q ! 1 (8.4) tends to the Gaussian distribution, i.e.expq.�ˇx2/ ! exp.�ˇx2/.

The above approach does not constitute, of course, the only possible choice foranalyzing the statistics of strongly correlated systems at thermal equilibrium. AsTsallis points out in his book [332], various entropic forms have been proposed bydifferent authors as alternatives to the BG entropy. Sq , however, turns out to enjoy anumber of important properties, also shared by SBG , that appear to render it superiorto other choices. For example, it satisfies uniqueness theorems analogous to those ofShannon and Khinchin obeyed by SBG and is Lesche stable (i.e. robust under smallvariations of the state probabilities pi ) and concave for all q > 0. By contrast, theRenyi entropy, for example, defined by [281]

SRq D k

lnPW

iD1 pqi

1 � q; (8.5)

8.2 Statistical Distributions of Chaotic QSS and Their Computation 195

although additive, fails to satisfy many other important requirements of entropicforms like concavity and Lesche stability [332].

What we wish to do in this chapter is study the complex statistics of several multi-dimensional Hamiltonian systems involving either nearest neighbor interactions orlong range forces. We focus on weakly chaotic regimes and demonstrate by meansof numerical experiments, in the spirit of the CLT, that pdfs of sums of orbitcomponents do not rapidly converge to a Gaussian, but are well approximated,for long integration times, by the q-Gaussian distribution (8.4). At longer times,of course, chaotic orbits generally leak out of smaller regions to larger chaotic seas,where obstruction by islands and cantori is less dominant and the dynamics is moreuniformly ergodic. This transition is signaled by the q-index of the distribution (8.4)decreasing towards q D 1, which represents the limit at which the pdf becomes aGaussian.

Thus, in our models, q-Gaussian distributions represent quasi-stationary states(QSS) that are often very long-lived, especially inside “thin” chaotic layers nearperiodic orbits that have just turned unstable. This suggests that it might be usefulto study these pdfs (as well as their associated q values) for sufficiently long timesand try to derive useful information regarding these QSS directly from the chaoticorbits, at least for time intervals accessible by numerical integration. QSS have alsobeen studied in coupled standard maps and the so-called Hamiltonian Mean Field(HMF) model in [22, 23], but not from the viewpoint of sum distributions.

One must be very careful, however, with regard to the kind of functions oneuses to approximate these pdfs. While it is true that q-Gaussians offer, in general,a quite accurate representation of QSS sum distributions, they are not the onlypossible choice. In fact, it has already been shown in certain systems that anotherso-called “crossover” function [77, 332, 333] can describe the same data with betteraccuracy. As pointed out in many references [108, 166, 167], there are cases whereother functions can be used to approximate better sum distributions of weaklychaotic QSS.

8.2 Statistical Distributions of Chaotic QSS and TheirComputation

Let us start again with an autonomous N dof Hamiltonian function of the form

H � H.q.t/; p.t// D H.q1.t/; : : : ; qN .t/; p1.t/; : : : ; pN .t// D E (8.6)

where .qk.t/, pk.t// are the positions and momenta respectively representing thesolutions in phase space at time t . What we wish to study here is the statisticalproperties of these solutions in regimes of weakly chaotic motion, where theLyapunov exponents [31, 32, 117, 310] are positive but very small. Such situations


arise often when one considers orbits which diffuse into thin chaotic layers andwander through a complicated network of “islands”, sticking for very long times tothe boundaries of these islands on a surface of constant energy.

There are several interesting questions one would like to ask here: How longdo these weakly chaotic states last? Assuming they are quasi-stationary, can wedescribe them statistically by observing some of their chaotic orbits? What type ofdistributions characterize these QSS and how could one connect them to the actualdynamics of the corresponding Hamiltonian system? Can we use these statisticalconsiderations to understand important physical properties of system (8.6) likeenergy equipartition? Is there any connection between the statistics of these QSSand the breakdown of certain localization phenomena discussed in earlier chapters?

The approach we shall follow is in the spirit of the well-known Central LimitTheorem [282] and is described in detail in [18]. In particular, Hamilton’s equationsof motion will be solved numerically for a large set of initial conditions to constructdistributions of suitably rescaled sums of M values of a generic observable �i D�.ti /, i D 1; : : : ; M , which depends linearly on the components of the solution.If these are viewed as different random variables (in the limit M ! 1), we mayevaluate their sum

S.j /M D

MX

iD1

�.j /i (8.7)

for j D 1; : : : ; Nic initial conditions. Thus, we can analyze the statistics of thesesums, centered about their mean value and rescaled by their standard deviation �M ,writing them as

s.j /M � 1

�M

�S

.j /M � hS.j /

M i�

D 1

�M

MX

iD1

�.j /i � 1

Nic

NicX

j D1

MX

iD1

�.j /i

!(8.8)

over the Nic initial conditions.

Plotting now the normalized histogram of the probabilities P.s.j /M / as a function

of s.j /M , we compare our pdfs with various functional forms found in the literature.

If the variables are independent and identically distributed, as explained in theprevious section, we expect a Gaussian. What we find, however, in many cases,is that the data is much better approximated by a q-Gaussian function of the form

P.s.j /M / D a expq.�ˇs

.j /2M / � a

�1 � .1 � q/ˇs

.j /2M

� 11�q

(8.9)

(see (8.4)). The normalization of this pdf is achieved by setting

ˇ D ap

��

3�q

2.q�1/

�

.q � 1/12 ��

1q�1

� ; (8.10)

8.3 FPU �-Mode Under Periodic Boundary Conditions 197

where � is the Euler � function, showing that the allowed values of q are1 < q < 3.

Let us now describe the numerical procedure one may use to calculate these pdfs.First, we specify an observable denoted by �.t/ as one (or a linear combination)of the components of the position vector q.t/ of a chaotic solution of Hamilton’sequations of motion, located initially at .q.0/; p.0//. Assuming that the orbit visitsall parts of a QSS during the integration interval 0 � t � tf, we divide this intervalinto Nic equally spaced, consecutive time windows, which are long enough tocontain a significant part of the orbit. Next, we subdivide each such window intoa number M of equally spaced subintervals and calculate the sum S

.j /M of the values

of the observable �.t/ at the right edges of these subintervals (see (8.7)). In this way,we treat the point at the beginning of every time window as a new initial conditionand repeat this process Nic times to obtain as many sums as required to have reliablestatistics. Consequently, at the end of the integration, we compute the average andstandard deviation of the sums (8.7), evaluate the Nic rescaled quantities s

.j /M and

plot the histogram P.s.j /M / of their distribution.

As we shall see in the next sections, in regions of weak chaos these distributionsare well-fitted by a q-Gaussian of the form (8.9) for fairly long time intervals.However, for longer times, the orbits often diffuse to wider domains of strong chaosand the well-known form of a Gaussian pdf is recovered.

8.3 FPU �-Mode Under Periodic Boundary Conditions

One very popular Hamiltonian to which we can apply our approach is the one-dimensional lattice of N particles with nearest neighbor interactions governed bythe FPU-ˇ Hamiltonian [121]

H.q; p/ D 1

2

NX

j D1

p2j C

NX

j D1

�1

2.qj C1 � qj /2 C 1

4ˇ.qj C1 � qj /4

�D E (8.11)

under periodic boundary conditions qN Cj .t/ D qj .t/, j D 1; : : : ; N , which wehave studied extensively in this book. More specifically, we shall first concentrateon orbits starting near a well-known NNM of this system called the �-mode, whichhas been studied in detail in several publications [16, 66, 67, 105, 218, 272]. Thissimple periodic solution is defined by (3.22)

qj .t/ D �qj C1.t/ � q.t/; j D 1; : : : ; N (8.12)

with N even.


10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j)

)P

(sM

(j))

P’(

s M(j)

)P

(sM

(j))

sM(j) sM

(j)

sM(j) sM

(j)

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 3010-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

a

c

b

d

Fig. 8.1 Plot in linear-log scale of numerical (solid curve), q-Gaussian (dashed curve) andGaussian (dotted curve) distributions for the FPU �-mode with periodic boundary conditions, withN D 128, ˇ D 1 and E D 0:768 > Eu � 0:0256. Panel (a) corresponds to final integrationtime tf D 105 using Nic D 104 time windows and M D 10 terms in the computation of thesums. Here, the numerical fitting with a q-Gaussian gives q � 1:818 with �2 � 0:0007. Panel (b)corresponds to tf D 106 , Nic D 104 and M D 100 and the numerical fitting gives q � 1:531 with�2 � 0:0004. Panel (c) corresponds to tf D 108 , Nic D 105 and M D 1; 000. It is evident that thenumerical distribution (solid curve) has almost converged to a Gaussian (dotted curve). Panel (d)compares the same pdf as in panel (a) with the QP function of (8.14) for a1 � 0:009, aq � 2:849

and q � 2:179 with �2 � 0:00008 (dashed curve) (after [18])

Our aim here is to investigate chaotic states near this orbit at energies where ithas just become unstable. To this end, let us choose as our observable the quantity

�.t/ D q N2

.t/ C q N2 �1.t/ (8.13)

which satisfies �.t/ D 0 at the �-mode. Thus, starting close to (8.12), �.t/ remainsnear zero at energies where the mode is stable and grows in magnitude at energieswhere the �-mode has destabilized, i.e. E > Eu. To compare with results publishedin the recent literature (see e.g. [218]), we first consider the case N D 128 andˇ D 1, for which Eu � 0:0257 [16] and take as our total energy E D 0:768

(i.e. " D E=N D 0:006), at which the �-mode is certainly unstable. The accuracyof the integration performed in [18] is determined at each time step by requiring thatH.q.t/; p.t// is within 10�5 from the energy value set initially at time t D 0.

As we see in Fig. 8.1, when the total integration time tf is increased, the pdfs(solid curves) approach closer and closer to a Gaussian with q tending to 1.


Moreover, this seems to be independent of the values of Nic and/or M , at leastup to the final integration time tf D 108. For example, when the parameters Nicand M in Fig. 8.1b are varied, one obtains q-Gaussians of very similar shape withq between 1.51 and 1.67. It is important to note, however, that the same data maybe better fitted by other similar looking functions: For example, in Fig. 8.1d thenumerical distribution (solid curve) is more accurately approximated by the so-called “crossover” function [77, 332, 333]

QP .s.j /M / D 1

n1 � aq

a1C aq

a1expŒ.q � 1/a1s

.j /2M �

o 1q�1

; a1; aq � 0 and q > 1; (8.14)

where a1 � 0:009, aq � 2:849 and q � 2:179 with �2 � 0:00008, in contrastto the �2 � 0:0007 obtained by fitting the same distribution by a q-Gaussian withq � 1:818 (see Fig. 8.1a). We note that �2 denotes the well known chi-square test ofreliability of a given statistical hypothesis, see e.g. [157]. Equation 8.14 representsa “crossover” between q-Gaussians and Gaussians and takes into account finitesize (and time) effects reflected in the lowering of the tails of the correspondingdistributions.

Regarding this QSS, it is interesting to note that when the final integration time isincreased beyond tf � 4 � 107, one observes that the LCEs monotonically increaseand attain bigger values than those computed up to tf ' 4�107 (see Fig. 8.2b). Thismay signify that the trajectories drift away from the neighborhood of the �-modeand enter a larger chaotic subspace of the energy manifold.

0

5

10

15

20

25

101 102 103 104 105 106 107 108 109

C0

t

10-5

10-4

10-3

10-2

10-1

LEs

t

101 102 103 104 105 106 107 108

a b

Fig. 8.2 (a) Plot of C0, (8.15), as a function of time for the unstable (E D 0:768 > Eu � 0:0257)�-mode with ˇ D 1 and N D 128. The grey curve corresponds to the time average of the solidcurve. (b) Log-log plot of the four biggest LCEs as a function of time for the same parameters asin panel (a) (after [18])


Similar results for this system have also been obtained by other authors using ananalogous methodology [218]. They too find that if one takes tf D 106 as the longestintegration time, one obtains pdfs that are well approximated by q-Gaussians. Theirtransitory character, however, is revealed when one integrates up to tf D 107 orlonger, when the pdfs are clearly seen to approach a Gaussian.

8.3.1 Chaotic Breathers and the FPU �-Mode

As we discovered in Chap. 6, physical properties like energy equipartition among alldegrees of freedom have been at the center of many investigations of FPU systems.In one such study, focusing on the �-mode under periodic boundary conditions[105], the authors studied the time evolution of an FPU-ˇ chain towards equipar-tition using initial conditions close to that mode. They observed that at energieswell above its threshold of destabilization, a remarkable localization phenomenonoccurs: A large amplitude excitation spontaneously occurs (strongly resembling adiscrete breather), which has a finite lifetime and moves chaotically along the chain,keeping all the energy restricted among very few particles. Moreover, numericalresults suggest that these “chaotic” breathers break down just before the systemreaches energy equipartition.

As pointed out in [105], this phenomenon can be monitored by evaluating thefunction

C0.t/ D N

PNnD1 E2

n

.PN

nD1 En/2; (8.15)

where En denotes the energy per site

En D 1

2p2

n C 1

2V.qnC1 � qn/ C 1

2V.qn � qn�1/ (8.16)

with 1 � n � N and V.x/ D 12x2 C ˇ

4x4. Since C0.t/ D 1, if En D E=N at each

site n and C0.t/ D N if the energy is localized at only one site, it can serve as anefficient indicator of energy localization in the chain.

In their experiments, these authors used N D 128, ˇ D 0:1, E � 42:2707 wellabove the lowest destabilization energy of the �-mode Eu � 0:25725 and plottedC0 versus t . Distributing evenly the energy among all sites of the �-mode at t D 0,they observed that C0 initially grows to relatively high values, indicating that theenergy localizes at a few sites. After a certain time, however, C0 reaches a maximumand decreases towards an analytically derived asymptotic value NC0 � 1:795 [105],which is associated with the breakdown of the chaotic breather and the onset ofenergy equipartition in the chain.


In [18], the same study of the �-mode was performed at a lower energy, E D0:768 > Eu � 0:0257 (keeping ˇ D 1, N D 128) and QSS were approximatedby q-Gaussian distributions, which are related to the lifetime of chaotic breathersas follows: In Fig. 8.2a, C0 is plotted as a function of time, verifying indeed thatit grows over a long interval (t � 1:8 � 108) after which it starts to decreaseand eventually tends to the asymptotic value NC0 ' 1:795 associated with energyequipartition. In Fig. 8.2b it is shown that the four biggest LCEs decrease towardszero until about t � 4 � 107, but then start to increase towards positive valuesindicating the unstable character of the �-mode.

In Fig. 8.3 we present the instantaneous energies En of all N D 128 sitesat different times, together with their associated sum distributions. As shown inFig. 8.3a, the energy becomes localized at only a few sites, demonstrating theoccurrence of a chaotic breather at times of order 104 . t . 108. Next, Fig. 8.3bshows that when the time is further increased (e.g. to t D 6 � 108), the chaoticbreather is destroyed and the system reaches equipartition.

Comparing Figs. 8.1, 8.2a and 8.3, we deduce that while the chaotic breather stillexists, a QSS is observed fitted by q-Gaussians with q well above unity, as seenin Fig. 8.3c. However, as the chaotic breather breaks down for t > 108 and energyequipartition is reached (see Fig. 8.3b), q ! 1 and q-Gaussian distributions rapidlyconverge to Gaussians (see Fig. 8.1c) in full agreement with what is expected fromBG statistical mechanics.

In Fig. 8.3d we plot estimates of the q values obtained, when one computeschaotic orbits near the �-mode at this energy density � D E=N D 0:006.Remarkably enough, even though these values have an error bar of about ˙10%due to the different statistical parameters M; Nic, used in the computation, theyexhibit a clear tendency to fall closer to 1 for t > 107, where energy equipartition isexpected to occur.

It is indeed a hard and open problem to determine exactly how equipartition timesTeq scale with the energy density � D E=N and other parameters (like ˇ), partic-ularly in the thermodynamic limit, even in one-dimensional FPU lattices. Althoughit is a question that has long been studied in the literature [25, 35, 37, 105, 111], theprecise scaling exponents by which Teq depends on �; ˇ, etc. are not yet preciselyknown for general mode excitations.

It would be very interesting if our q values could help in this direction. In fact,carrying out more careful calculations, as in Fig. 8.3d, at other values of the specificenergy, e.g. � D 0:04 (see Fig. 8.4) and � D 0:2, one finds q plots that exhibita clear decrease to values close to 1, at Teq � 7:5 � 105 and Teq � 7:5 � 104

respectively, approximately where the corresponding chaotic breathers collapse.Still, even though these results are consistent with what is known in the literature,the limited accuracy of the above approach does not allow one to say somethingmeaningful about scaling laws, as the system tends to equilibrium.


0

0.05

0.1

0.15

0.2

0.25

1 20 40 60 80 100 120

En

n

0

0.005

0.01

0.015

0.02

0.025

1 20 40 60 80 100 120

En

n

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j)

)

sM(j)

0

0.5

1

1.5

2

2.5

3

105 106 107 108

q-en

trop

ic p

aram

eter

t

a

c d

b

Fig. 8.3 In panel (a) at t D 107 , near the maximum of C0.t/ (see Fig. 8.2a), we see a chaoticbreather. In panel (b) at t D 6 � 108, this breather has collapsed and the system has reached a statewhose distribution is very close to a Gaussian, as implied already by the pdf shown in Fig. 8.1c att D 108 . Note the scale difference in the vertical axes. Panel (c) at t D 107 shows that the sumdistribution, near the maximum of the chaotic breather, is still quite close to a q-Gaussian withq � 2:6. Panel (d) presents an estimate of the q index at different times, which shows that itsvalues on the average fall significantly closer to 1 for t > 107 (after [18])

0

0.5

1

1.5

2

2.5

3

0 2.5 106 5 106 7.5 106 107

q-en

trop

ic p

aram

eter

t

Fig. 8.4 Plot of q values as afunction of time for energydensity � D 0:04, i.e.E D 5:12 (dashed curve) forthe �-mode of an FPUperiodic chain with N D 128

particles and ˇ D 1. Thedotted and solid curves showerror bars in the form of plusand minus one standarddeviation. In agreement withother studies the transition tovalues close to q D 1 occurshere at Teq � 7:5 � 105

(after [18])

8.4 FPU SPO1 and SPO2 Modes Under Fixed Boundary Conditions 203

1.6

1.8

2

2.2

2.4

2.6

2.8

-0.2 -0.1 0 0.1 0.2

p 1

q1

Fig. 8.5 The “figure eight” chaotic region (blue color) is observed for an initial condition at adistance close to the unstable SPO1 mode (depicted as the saddle point at q3 D 0 and p3 > 0).A slightly more extended “figure eight” region (green points) occurs for an initial condition a littlefurther away and a large scale chaotic region (red points) arises for an initial condition even moredistant on the surface of section (q1; p1) computed at times when q3 D 0. Orbits are integrated upto tf D 105 using E D 7:4, N D 5 and ˇ D 1:04 (after [18])

8.4 FPU SPO1 and SPO2 Modes Under Fixed BoundaryConditions

Let us now examine the chaotic dynamics near NNMs of the FPU system underfixed boundary conditions, i.e. q0.t/ D qN C1.t/ D 0. In particular, we first evaluatepdfs of sums of chaotic orbit components near the SPO1 mode (see (3.26)), whichkeeps one particle fixed for every two adjacent particles oscillating with oppositephase. This mode is defined for N odd by

q2j .t/ D 0; q2j �1.t/ D �q2j C1.t/; j D 1; : : : ;N � 1

2: (8.17)

As shown in Fig. 8.5, the chaotic region close to this solution (when it has justbecome unstable) appears for a long time isolated in phase space from other chaoticdomains. In fact, one finds several such domains, embedded one into each other.For example, in the case N D 5 and ˇ D 1:04, a “figure eight” chaotic regionappears on the surface of section (q1; p1) of Fig. 8.5 computed at times when q3 D 0

(and p3 > 0) at the energy E D 7:4. Even though the SPO1 mode is unstable(depicted as the saddle point at the middle of this surface of section) orbits startingsufficiently nearby remain in its vicinity for very long times, forming eventually thethin blue “figure eight” at the center of the figure. Starting, however, at points a littlefurther away a more extended chaotic region is observed, plotted by green points,which still resembles a “figure eight”. Choosing even more distant initial conditions,a large scale chaotic region plotted by red points becomes evident in Fig. 8.5.


It is, therefore, reasonable to regard these different dynamical behaviors near theSPO1 mode as QSS and characterize them by pdfs of sum distributions, as explainedin Sect. 8.2. The idea behind this is that orbits starting initially at the immediatevicinity of the unstable SPO1 mode behave differently than those lying further away,since the latter orbits have the ability to explore more uniformly parts of the constantenergy surface. Thus, q-Gaussian-like distributions with 1 < q < 3 are expectednear SPO1, while for orbits starting sufficiently far the distributions we expect tofind are Gaussians with q ! 1.

To test the validity of these ideas, let us choose the quantity

�.t/ D q1.t/ C q3.t/ (8.18)

as our observable, which is exactly equal to zero at the SPO1 orbit. Following whatwas presented in Sect. 8.2, we now study the motion related to three different initialconditions as a result of integration over longer and longer times, during which anorbit passes through all the different stages depicted in Fig. 8.5. In particular, inFig. 8.6a, we see the surface of section created by the trajectory starting close tothe NNM and integrated up to tf D 105 while in the following two panels we seethe same surface of section computed for final integration times of tf D 107 andtf D 108 respectively. The parameters are the same as in Fig. 8.5.

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

p 1

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

p 1

q1

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

q1

10-5

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j)

)

sM(j)

10-5

10-4

10-3

10-2

10-1

100

P(s

M(j)

)

P(s

M(j)

)

-30 -20 -10 0 10 20 30

sM(j)

10-5

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

sM(j)

a

d e f

b c

-4

-3

-2

-1

0

1

2

3

4

-2 -1 0 1 2

p 1

q1

Fig. 8.6 (a) The .q1; p1/ surface of section of anf orbit integrated up to tf D 105 and startingclose to the unstable SPO1 orbit for N D 5 and ˇ D 1:04 at energy E D 7:4. (b) and (c) are sameas (a) but for tf D 107 and tf D 108 respectively. (d)–(f) Plots in linear-log scale of numerical(solid curve), of q-Gaussian (dashed curve) and Gaussian (dotted curve) for th einitial conditionsof (a)–(c) respectively. In particular: (d) is for tf D 105, Nic D 104 and M D 10, (e) for tf D 107,Nic D 105 and M D 1; 000 terms, and (f) for tf D 108 , Nic D 105 and M D 1; 000 (after [18])


Clearly, as the integration time increases, orbits starting close to the unstableSPO1 mode, eventually wander over a more extended part of phase space, coveringgradually all of the energy surface when the integration time is sufficiently large(e.g. tf D 108).

An important question arises here: Are these behaviors reflected in the statisticsassociated with these trajectories? The answer to this question is presented inFig. 8.6d–f. In particular, Fig. 8.6d shows in linear-log scale the numerical (solidcurve), q-Gaussian (dashed curve) and Gaussian (dotted curve) distributions for theinitial condition located closest to SPO1, using tf D 105, Nic D 104 time windowsand M D 10 terms in the computations of the sums. In this case, a q-Gaussian fittingof the data gives q � 2:785 with �2 � 0:000 31. This distribution corresponds tothe surface of section shown in Fig. 8.6a. If one now increases tf by two orders ofmagnitude (see panel (f)) using Nic D 105, M D 1; 000 and performs the samekind of fit one gets q � 2:483 with �2 � 0:000 47.

It is important to emphasize, however, that the lower parts (tails) of the solidcurve distribution of Fig. 8.6e are not fitted well by a q-Gaussian. This suggeststhat by increasing the integration time, the initial pdf takes a form that may wellbe approximated by other types of functions, like e.g. (8.14). This distributioncorresponds to the surface of section of Fig. 8.6b. By increasing the time furtherto tf D 108 and using Nic D 105 and M D 1; 000 terms, we observe that the solidcurve of panel (g) is indeed very close to a Gaussian (q � 1:05), characterizing thechaotic regime plotted in the surface of section of Fig. 8.6c.

Next, the authors of [18] turned to another nonlinear mode of the FPU Hamilto-nian with fixed boundary conditions called the SPO2 mode (see (3.27)). This is aNNM which keeps every third particle fixed, while the two in between move in exactout-of-phase motion. What is important about this NNM is that it becomes unstableat much lower energies (i.e. Eu=N / N �2) compared to SPO1 (Eu=N / N �1)[13], much like the low k D 1; 2; 3; : : : mode periodic orbits connected with thebreakdown of FPU recurrences [64, 94]. Thus, it is expected that near SPO2, orbitswill be more weakly chaotic than SPO1 and hence QSS should persist for longertimes. This is exactly what happens. As Fig. 8.7 clearly shows, the dynamics in aclose vicinity of SPO2 has the features of what we might call “edge of chaos”:Orbits wander in a regime of very small (positive) LCEs, tracing a kind of “banana”-shaped region much different than the “figure eight” we had observed near SPO1.Remarkably, the pdfs in this case (at least up to tf D 1010), actually converge to asmooth function, never deviating towards a Gaussian, as in the QSS of other FPUsystems.

More specifically, let us set N D 5, ˇ D 1, E D 0:5 and choose an orbitlocated initially close to the SPO2 solution, which has just turned unstable (at Eu �0:4776). As we can see in Fig. 8.7a, the dynamics here yields banana-like orbits atleast up to tf D 1010. The weakly chaotic nature of the motion is plainly depictedin Fig. 8.7b, where the four positive LCEs are plotted. Note that, although they dodecrease towards zero for a very long time, at about tf � 109, the largest one ofthem tends to converge to a very small value (about 10�8), indicating that the orbitis chaotic, sticking perhaps to an “edge of chaos” region around SPO2.


-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

p 1

q1

10-10

10-8

10-6

10-4

10-2

100

101

102

103

104

105

106

107

108

109

1010

LEs

t

10-5

10-4

10-3

10-2

10-1

100

10-5

10-4

10-3

10-2

10-1

100

-40 -30 -20 -10 0 10 20 30 40

P(s

M(j)

)

sM(j)

-40 -30 -20 -10 0 10 20 30 40

sM(j)

P’(s

M(j)

)

a b

c d

Fig. 8.7 (a) The .q1; p1/ surface of section of an orbit integrated up to tf D 1010 starting inthe vicinity of the unstable SPO2 mode. (b) The corresponding four biggest LCEs. (c) Linear-logscale plot of the numerical (solid curve), q-Gaussian (dashed curve) and Gaussian (dotted curve)distributions. (d) The solid curve distribution of panel (c) is better fitted by the dashed curve of theQP function of (8.14) (after [18])

In Fig. 8.7c, the corresponding pdf is plotted at time tf D 1010 (whose shapedoes not change after tf D 107). An extremely long-lasting QSS is formed, whosedistribution is well-fitted by a q-Gaussian with q � 2:769 and �2 � 4:44 � 10�5.The “legs” of this distribution away from the center deviate from the q-Gaussianshape, but remain far from the Gaussian plotted as a dotted curve in the figure.Performing a similar fitting of our data with the function (8.14), in Fig. 8.7d, as in thecase of Fig. 8.1d, one finds that the numerical distribution (solid curve) of Fig. 8.7cis better approximated by (8.14) where a1 � 0:006, aq � 170 and q � 2:82 with�2 � 2:06 � 10�6, compared with the �2 � 4:44 � 10�5 obtained by fitting thesame distribution by a q-Gaussian with q � 2:769 in Fig. 8.7c.

Thus, in “thin” chaotic layers of multi-dimensional Hamiltonian systems withsmall positive LCEs it is possible to find non-Gaussian QSS that persist for verylong times as in the SPO2 case. Numerical evidence suggests that in these regimeschaotic orbits stick for long time intervals to a complex network of islands, where


0

0.5

1

1.5

2

2.5

3

3.5

4

101

102

103

104

105

106

107

108

C0

t

-30 -20 -10 0 10 20 300.001

0.01

0.1

P(s

M(j)

)

0.001

0.01

0.1

P(s

M(j)

)

sM(j) sM

(j)-40 -30 -20 -10 0 10 20 30 40

10-4

10-3

10-2

10-1

LEs

t

101

102

103

104

105

106

107

108

a b

c d

Fig. 8.8 (a) Plot of C0 (8.15) (solid curve) as a function of time for a perturbation of the unstable(E D 1:5 > Eu � 1:05226) SPO1 mode with ˇ D 1:04 and N D 129. The grey curvecorresponds to the time average of the solid curve. (b) Log-log plot of the four biggest LCEs asa function of time. Plots in linear-log scale of numerical (solid curve), q-Gaussian (dashed curve)and Gaussian (dotted curve) distributions: (c) for final integration time tf D 4 � 105 , Nic D 104

time windows and M D 20 terms in the sums, (d) for tf D 2 � 106, Nic D 5 � 104 and M D 20

(after [18])

their statistics is well approximated by q-Gaussian distributions connected withnonextensive statistical mechanics.

It is interesting to study the dynamics near the unstable SPO1 and SPO2 modesusing an analysis similar to the one carried out for the �-mode, in Sect. 8.3.1,following [105]. In particular, focusing on small perturbations of both modes,one may wish to investigate how C0 (8.15) depicts the evolution towards energyequipartition and whether this transition will again be preceded by the appearanceof chaotic breathers.

To find out, consider the SPO1 mode and follow neighboring orbits starting at asmall distance from the mode when it has just turned unstable. As explained in detailin [18], C0 is found to grow on the average without a clear maximum, settling downto a value near 1:75 (see Fig. 8.8), indicating that the system reaches equipartitionat about t � 106, where the four biggest LCEs begin to converge to their finalvalues, as shown in Fig. 8.8b. Interestingly, during this period, the corresponding


distribution in Fig. 8.8c (using the observable � D q.64/ C q.65/ C q.66/) is wellfitted by a q-Gaussian with q � 2:843 at t D 4�105. For longer times, distributionsquickly tend to Gaussians as shown in Fig. 8.8d for t D 2 � 106. Thus, even though(unlike the �-mode) no chaotic breather is observed in the time C0 takes to relax toits limiting value, sum distributions are well approximated by q-Gaussians and onlyconverge to Gaussians when energy equipartition has occurred.

Repeating this process now for an orbit starting close to the SPO2 mode, whenit has just destabilized, one finds that C0 also does not exhibit a distinct highmaximum, but grows on the average more slowly than in the SPO1 case (see[18]). Thus, it takes longer to approach its limit, indicating that the system reachesequipartition at t � 4 � 108, where the four biggest Lyapunov exponents cease todecrease towards zero and tend to small positive values. The corresponding sumdistribution is well fitted by a q-Gaussian with a high q D 1:943 value, while weneed to increase the time to 5 � 108 to see pdfs that are much closer to a Gaussian(q � 1:0), indicating that this is a lower bound for energy equipartition.

8.5 q-Gaussian Distributions for a Small MicroplasmaSystem

We now turn our attention to a Hamiltonian system very different than the FPUsystems of the previous sections and discuss a number of results presented in[18], concerning a system of few degrees of freedom characterized by long rangeinteractions of the Coulomb type. As we will see, the approach described in thischapter will allow us to study statistically the dynamics of its transition from acrystal-like to a liquid-like phase (the so-called “melting transition”) [17, 168] atsmall energies, as well as identify a transition from the liquid-like to a gas-likephase, as the total energy increases further [145].

We shall focus, in particular, on a microplasma system consisting of N ions ofequal mass m D 1 and electric charge Q moving in a Penning trap in the presenceof an electrostatic potential [17, 145]

˚.x; y; z/ D V0

2z2 � x2 � y2

r20 C 2z2

0

(8.19)

(r0; z0 are physical parameters of the trap), and a constant magnetic field in the zdirection, whose vector potential is

A.x; y; z/ D 1

2.�By; Bx; 0/: (8.20)

8.5 q-Gaussian Distributions for a Small Microplasma System 209

The system is described by the Hamiltonian

H DNX

iD1

(1

2mŒpi � qA.ri /�

2 C Q˚.ri /

)C

X

1�i<j �N

Q2

4�"0rij

; (8.21)

where ri is the position of the i th ion, rij is the Euclidean distance between i thand j th ions and "0 is the vacuum permittivity. In the Penning trap, the ions aresubjected to a harmonic confinement in the z direction with frequency

!z D"

4QV0

m.r20 C 2z2

0/

# 12

(8.22)

while, in the perpendicular direction they rotate (due to the cyclotron motion)with frequency !c D QB=m. Thus, in a frame rotating about the z axis withLarmor frequency !L D !c=2, the ions are subjected to an overall harmonicpotential with frequency !x D !y D .!2

c=4 � !2z =2/1=2 in the direction

perpendicular to the magnetic field. In the rescaled time D !ct , position R D r=a

and energy H D H =.m!2ca2/ with a D ŒQ2=.4�"0m!2

c/�1=3, our Hamiltonian(8.21) reads

H D 1

2

NX

iD1

P2i C

NX

iD1

h�1

8� 2

4

�.X2

i C Y 2i / C 2

2Z2

i

iCX

i<j

1

Rij

D E (8.23)

where E is the total constant energy, Ri D .Xi ; Yi ; Zi / and Pi D .PXi ; PYi ; PZi /

are the positions and momenta in R3 respectively of the N ions, Rij is the Euclidean

distance between different ions i; j given by

Rij D Œ.Xi � Xj /2 C .Yi � Yj /2 C .Zi � Zj /2�12 (8.24)

and D !z=!c.Due to the form of the potential in (8.23), the ions perform bounded motion

provided j j < 1=p

2. The Penning trap is called prolate if j j < 1=p

6, isotropicif j j D 1=

p6 and oblate if 1=

p6 < j j < 1=

p2. Thus, the motion is quasi one-

dimensional in the limit ! 0 and quasi two-dimensional in the limit ! 1=p

2.The Z direction is a symmetry axis and hence, the Z component of the angularmomentum, LZ D PN

iD1 XiPYi � Yi PXi , is conserved, being a second integral ofthe motion. We may, therefore, set from here on the angular momentum equal tozero (i.e. LZ D 0) and study the motion in the Larmor rotating frame.

In [17], the authors demonstrate the occurrence of a dynamical regime changein a microplasma system composed of N D 5 ions and confined in a prolate quasione-dimensional configuration of D 0:07. More specifically, in the lower energyregime, a transition from crystal-like to liquid-like behavior was observed, called the


0

0.5

1

1.5

2

2.5

3

2 3 4 5 6 7 8 9 10

q-en

trop

ic p

aram

eter

E

Fig. 8.9 Plot in log-linearscale of the q parameter (solidcurve) as a function of theenergy E of the microplasma(8.23) for D 0:07 (prolatetrap) and N D 5. Plotted hereis also the line q D 1 forcomparison with theGaussian case (after [18])

“melting” phase transition. First, it was shown that this transition is not associatedwith a sharp increase of the temperature at some critical energy, as might have beenexpected at first sight. Furthermore, the positive LCEs achieve their maximum atenergies much higher than the “melting” regime. Thus, it appears that no clearmacroscopic reasoning is available for identifying and analyzing this process.

For this reason, the SALI method [309, 313, 314] described in Chap. 5 wasinvoked to study the local microscopic dynamics in detail [17]. It was discoveredthat there exists indeed an energy range of weakly chaotic behavior, i.e. E 2�Emt D .2; 2:5/, where the positive LCEs are very small and the SALI exhibitsa stair-like decay to zero with varying decay rates. This suggests the presence oflong-lived “sticky” components executing a multi-stage diffusion process near theboundaries of resonance islands [316]. Thus, it was concluded that it is in this energyinterval that the above “melting” transition occurs.

Motivated by these results, the investigation was pursued further in [18], forthe same and N , in the regime where the minimum energy at which ions startmoving appreciably about their equilibria positions is E0 � 1:8922. The aim wasto study the melting transition as the energy E of the system increases above E0,using pdfs associated with chaotic trajectories to relate microscopic to macroscopicobservables. Based on the results described earlier in this chapter, one might expectto find q-Gaussian approximations with 1 < q < 3 in the vicinity of �Emt, wherethe positive LCEs Li , i D 1; : : : ; 3N are quite small compared to their maximumvalue of L1 � 0:0558 attained at E � 5:95.

Figure 8.9 shows the results of this study for the interval .E0; 10�. Choosing asan observable the quantity �.t/ D X1.t/ and setting Nic D 2 � 104, M D 1; 000

the equations of motion were integrated for a total time tf D 2 � 107. Remarkably,in the energy range �Emt of the “melting” transition, the values of the entropicindex of the q-Gaussian pdfs approximating the data, were found to be well aboveq D 1, indicating that the statistics is certainly not Gaussian. In fact, the detected

8.5 q-Gaussian Distributions for a Small Microplasma System 211

10-6

10-5

10-4

10-3

10-2

10-1

2 5 10 30 50 100 200 300

λ 1, λ

2

E

0

0.5

1

1.5

2

2.5

3

10 30 50 100 200 300

q-en

trop

ic p

aram

eter

E

a b

Fig. 8.10 (a) The two biggest positive LCEs (L1 (solid curve) and L2 (dashed curve), denoted by�1 and �2 respectively. as a function of the energy E in log-log scale. (b) Plot in log-linear scale ofthe q parameter (solid curve) as a function of the energy E of the microplasma Hamiltonian (8.23)for D 0:07 and N D 5. Plotted here is also the line q D 1 for comparison with the Gaussiancase (after [18])

q-Gaussian shape persists up to E � 4, implying that the energy interval of thetransition may be slightly bigger than was originally estimated.

Next, in [18] the second transition of system (8.23) was studied, from a liquid-like to a gas-like phase. Here, the system is studied over an energy range wherethe largest positive LCEs decrease towards zero following (8.25), as shown inFig. 8.10a. Thus, as one moves to higher energies, the system passes from a stronglychaotic regime at energies where the LCEs are maximal (i.e. for E & 6) to energieswhere the motion is much less chaotic.

As demonstrated in [145], if all N 2 inter-particle interaction terms of theCoulomb part of Hamiltonian (8.23) at distances Rij contribute to the MLE L1,it is possible to derive, for sufficiently large values of T ! 1 (and therefore ofE ! 1), that

L1 *

N 2

R3ij

+ N

.ln T /12

T34

(8.25)

where T is related to the temperature of the system. This formula explains theasymptotic power law decay of the biggest LCE observed in Fig. 8.10a. It can alsobe shown that the second largest LCE L2 (dashed curve) obeys a similar formula asa function of E .

The important result obtained in [18] is that, in this case also, the q indices ofthe corresponding pdfs remain well above unity for E 2 .30; 200/, as shown inFig. 8.10b. This suggests that in this range also a dynamical change to a weakerform of chaos occurs, associated with small positive LCEs. Most probably, the N 2

inter-particle terms do not contribute significantly to the transition between the fullydeveloped chaotic regime of a liquid-like state and the more ordered dynamics of agas-like regime.


It is indeed interesting to compare the plots of the index q as a function of energyshown in Figs. 8.9 and 8.10b, with the behavior of q as a function of time shown inFigs. 8.3d and 8.4 for the chaotic regime near the �-mode of the FPU system. Theseresults demonstrate that, in very different situations, studying the statistics of chaoticQSS and monitoring the values of the entropic index of q-Gaussian approximationscan prove very useful in revealing important physical phenomena like the onset ofenergy equipartition, or the occurrence of dynamical phase transitions.

8.6 Chaotic Quasi-stationary States in Area-Preserving Maps

Two-dimensional area-preserving maps are excellent models for studying the qual-itative dynamics of Hamiltonian systems of two degrees of freedom. The Poincaremaps that we extensively analyzed in Chaps. 2 and 3 are all area-preserving maps.As such, they generically possess families of invariant closed curves (representingquasiperiodic orbits), which form complete barriers to all motion evolving inside“islands” centered around stable periodic orbits in the two-dimensional phasespace. At the boundaries of these islands, a complex network of smaller islandsand invariant Cantor sets (often called cantori) exists, to which chaotic orbits areobserved to “stick” for very long times. It is here that trajectories often get trappedinto QSS that can be very long-lived. Such phenomena have so far been thoroughlystudied in terms of a number of dynamical mechanisms responsible for chaotictransport (see e.g. [236, 238, 347]).

It would, therefore, be very natural to wonder: Could we also study the dynam-ics in these “sticky” regions using the probabilistic techniques of nonextensivestatistical mechanics? What would pdfs of sums of iterates reveal that wouldhelp us understand the complexity of motion in these domains? Will we uncoverinteresting connections between the “geometry” of phase space dynamics and thetime-evolving statistics of chaotic orbits, as we did in earlier sections through asimilar analysis of multi-dimensional Hamiltonian systems? As we describe in whatfollows, it is indeed possible to find in such “weakly chaotic” domains of area-preserving maps long-lived QSS, whose pdfs do not rapidly converge to a Gaussian,but pass through several stages, some of which are very well approximated byq-Gaussian distributions.

Let us start by recalling a number of studies of such pdfs in 1D maps [7, 292,327, 328], as well as higher-dimensional conservative maps [115], in similar “edgeof chaos” domains, where the MLE either vanishes or is very close to zero. Thesestudies have already provided ample evidence of the usefulness of q-Gaussiandistributions, in the context of the CLT. Despite the controversy that some of thesefindings have generated [68, 158], it is worth pointing out that for one-dimensionalmaps, the situation is a lot clearer. Indeed, as shown convincingly in [328, 333],when one approaches the critical point of the period-doubling route to chaos takinginto account a proper scaling relation that involves the Feigenbaum constant ı and

8.6 Chaotic Quasi-stationary States in Area-Preserving Maps 213

the location of the critical point, the pdfs of sums of iterates of the logistic mapare approximated by a q-Gaussian far better than the Levy distribution proposed in[158]. This suggests the need for a more thorough investigation of low-dimensionalmaps, within a nonextensive statistical mechanics approach, in which q-Gaussiandistributions represent metastable states, or QSS of the dynamics [22,23,251,285].

Since we are going to work in the context of the CLT, it is important to note thatsuch a theorem has been verified for deterministic systems [29,44,237]. Attempts togeneralize the CLT have also been published demonstrating that, for certain classesof strongly correlated random variables, their rescaled sums approach a q-Gaussianlimit distribution [161, 335, 336]. As stated earlier, however, objections have alsobeen raised by some authors [108, 166, 167], questioning whether q-Gaussians canindeed constitute attractors, as in some mathematical models where this was thoughtto be the case, other functions were shown to describe the sum distributions in theCLT limit.

To understand the situation better, therefore, it is instructive to probe deeperinto the complex dynamics of conservative maps in weakly chaotic domains andinvestigate pdfs of rescaled sum of M iterates, in the large M limit, and for manydifferent initial conditions. We will thus be able to shed more light on possibleconnections between “geometric” properties of chaotic regions and their statisticsexpressed by pdfs of long-lived QSS in these domains. As shown recently in [293],we will discover that, in general, as M grows, these pdfs pass from a q-Gaussianto an exponential form (having a triangular shape in semi-log plots), ultimatelytending to become true Gaussians, as “stickiness” to an ‘edge of chaos’ subsidesin favor of more uniformly chaotic motion. Still, we will also demonstrate that thereare cases where the orbits evolve in such convoluted pathways that q-Gaussianapproximations persist for as long as it was possible to iterate the equations ofmotion.

8.6.1 Time-Evolving Statistics of pdfs in Area-Preserving Maps

Let us, therefore, consider in what follows 2D maps of the form

xnC1 D f .xn; yn/; ynC1 D g.xn; yn/ (8.26)

and treat their chaotic orbits as generators of random variables, as we did forHamiltonian systems earlier in this chapter. We will thus compute long sequencesproduced by many initial conditions and examine whether these can be consideredas independent and identically distributed quantities as required by the classicalCLT. In this regard, it is important to recall that the well known CLT assumptionabout the independence of identically distributed random variables can be replacedby the weaker property of asymptotic statistical independence, as shown in [237].


Thus, we may proceed to compute the generalized rescaled sums of one of theiterates, say xn of the map (8.26) in the sense of the CLT [29, 44, 237]:

ZM D M �

MX

nD1

.xn � hxi/ (8.27)

where hi implies averaging over a large number of iterations M and a large numberof randomly chosen initial conditions Nic. For fully chaotic systems D 1=2 andthe distribution of (8.27) in the limit (M ! 1) is expected to be a Gaussian [237].Alternatively, however, we may define the non-rescaled variable

zM DMX

nD1

Œxn � hxi� (8.28)

(absorbing the factor M � ) and analyze its pdf normalized by its variance as follows(see [293] and Sect. 8.2 of this chapter): First, we construct the sums S

.j /M obtained

by adding iterates xn .n D 0; : : : ; M / of the map (8.26), where .j / represents thedependence on the randomly chosen initial conditions x

.j /0 , with j D 1; 2; : : : ; Nic.

Next, we focus on the sjm variables, which are the S

jms centered about their mean and

rescaled by their standard deviation � , and follow the procedure outlined in Sect. 8.2(see (8.8)).

More specifically, we compute the pdfs of s.j /M and plot, in the sections that

follow, the corresponding histograms of P.s.j /M / for sufficiently small increments

(e.g. �s.j /M D 0:05) to smoothen out fine details and analyze their functional form.

Following [293], we express in our plots the independent variable as z=� � s.j /M .

Before proceeding, however, to investigate in detail these phenomena in a familyof area-preserving maps, we refer the reader to Problem 8.1 below, where he (or she)is asked to apply the above methodology to a 2D map, which possesses a strangeattractor when it is dissipative and an annular region with complex chaotic dynamicswhen it is area-preserving. It is indeed remarkable how the Gaussian that dominatesthe statistics in the presence of a strange attractor, yields its place to a differentfunction, which is well-approximated by a q-Gaussian in the conservative case.

8.6.2 The Perturbed MacMillan Map

Let us consider the so-called perturbed MacMillan map, introduced in [151] to studythe onset of chaos via Mel’nikov theory, in the form:

8<

:xnC1 D yn

ynC1 D �xn C 2 yn

1 C y2n

C " .yn C ˇxn/(8.29)


where ", ˇ, are physically important parameters. For " D 0, (8.29) is knownto be integrable possessing a simple polynomial invariant that allows it to have asaddle point at the origin, for > 1 (see Exercise 8.1). The Jacobian determinant isJ D 1 � "ˇ, so that (8.29) is area-preserving for ˇ D 0, and dissipative for "ˇ > 0.Here, we only consider the area-preserving case ˇ D 0, so that the only relevantparameters are .", /.

As discussed in Exercise 8.1, for > 1 the unperturbed map possesses a “figureeight” invariant curve with a self-intersection at the origin that is a fixed point ofsaddle type. For " > 0, the invariant manifolds of this saddle split and a thin chaoticlayer appears surrounding two large islands in the .xn; yn/ plane. The MLE L1

for D 1:6 and 0:2 � � � 1:8 is found to vary between 0:0875 and 0:03446.By analyzing the histogram of the normalized sums of (8.28) within this chaoticlayer for a wide range of parameters " and , we find that, in many cases, the pdfsbegin by being well approximated by q-Gaussians and then turn to exponentials e�kjzj, whose triangular shape on semi-log plots eventually starts to approachan inverted parabola representing the Gaussian function. Thus, monitoring chaoticorbits in this region for increasingly large numbers of iterations M , we observethe occurrence of different QSS described by these distributions. One thus obtainssome very interesting results, which are described in detail in [293] and are brieflydiscussed in the sections that follow.

Let us focus, in particular, on the time-evolving statistics of two examples of theMacMillan map, which represent respectively: (1) One set of cases with a “figureeight” chaotic domain whose distributions pass through a succession of pfds beforeconverging to an ordinary Gaussian, and (2) a set with more complicated domainsextending around many islands, where q-Gaussian pdfs dominate the statistics forvery long times and no tendency to a Gaussian is observed.

8.6.2.1 The " D 0:9, � D 1:6 Class of Examples

The case " D 0:9, D 1:6 is a typical example characterized by time-evolvingpdfs. As shown in Fig. 8.11b, the corresponding phase space plot yield a seeminglysimple chaotic region in the form of a “figure eight”, while the corresponding pdfsdo not converge to a single distribution, passing from a q-Gaussian-looking functionto an exponential distribution.

Analyzing carefully the time evolution of chaotic orbits in this example, weobserve that there exist at least three long-lived QSS. In particular, for i D1; : : : ; M D 216, a QSS is produced whose pdf is close to a pure q D 1:6-Gaussian. Figure 8.12 shows some pdfs for different numbers of iterates M . Notethat these pdfs correspond to a “figure eight” chaotic region that evolves essentiallyaround two large islands (Fig. 8.11(a)). However, for M > 216, a more complexstructure emerges: Iterates stick around new islands, and a transition is evident fromq-Gaussian to exponentially decaying pdfs [293].


Fig. 8.11 (a) pdfs of the renormalized sums of M iterates of the MacMillan map (8.29), for " D0:9, D 1:6, M � 216 and Nic initial conditions randomly chosen within a square .0; 10�6/ �.0; 10�6/ about the origin. (b) Phase space plot for M D 216 (after [293])

MM

M

M

M

M

M

M

Fig. 8.12 Detailed evolution of the pdfs of the MacMillan map for " D 0:9, D 1:6, as M

increases from 212 to 226, respectively (after [293])

Clearly, therefore, for " D 0:9 (and other cases with " D 0:2, 1.8) more than oneQSS coexist whose pdfs are well-approximated by a sequence of q-Gaussians. Infact, for 1018 � M � 221, the central part of the distribution is still well-fitted bya q-Gaussian with q D 1:6. However, as we continue to iterate the map to M D223, intermediate states are observed, which are better described by an exponentialdistribution. From here on, as M > 223, the central part of the pdfs is close to aGaussian (see Fig. 8.12) and a true Gaussian is expected in the limit (M ! 1).The evolution of this sequence of successive QSS as M increases is shown in detailin Fig. 8.12.


4

4

4

6

4

2

yn

Xn

Xn

Xn Xn

Xn

Xn

yn

yn yn

yn

yn

2

2 M=216

M=29 M=213

M=217

M=2184

2

642

2

-2

-2

-2

-2-4

4

2

-2

-4

-4

-2

-2

-4

-4

-6

6

4

2

-2

-4

-6

6

4

2

-2

-4

-6

-6 642-2-4-6

642-2-4-6-4

-4 42-2-4

219 £ M £ 220

Fig. 8.13 Structure of phase space plots of the MacMillan map (8.29) for parameter values " D1:2 and D 1:6, starting from a random set of initial conditions chosen randomly within a square.0; 10�6/ � .0; 10�6/ about the origin and iterating the map M times (after [293])

8.6.2.2 The � D 1:2, � D 1:6 Class of Examples

Let us now turn to the behavior of the " D 1:2, D 1:6 class, whose MLE isL1 � 0:05, hence smaller than that of the " D 0:9 case (where L1 � 0:08). Asclearly seen in Fig. 8.13, a diffusive behavior sets in here that extends outward inphase space, enveloping a chain of eight large islands to which the orbits “stick” asthe number of iterations grows to M D 219.

Some representative pdfs of this evolution are seen in Fig. 8.14. In this classof MacMillan maps, the chaotic domain under study extends around several largeislands and is apparently richer in “stickiness” phenomena. This higher complexityof the dynamics may very well be the reason why the corresponding chaotic states


Fig. 8.14 Pdfs of therenormalized sums of M

iterates of the MacMillanmap (8.29), for " D 1:2, D 1:6, M D 218, 219 and220. Note the apparentconvergence to a q-Gaussian,due to the complicatedchaotic dynamics around thelarge islands of Fig. 8.13(after [293])

possess pdfs that are well approximated by q-Gaussians with q > 1 and persist forextremely large numbers of iterations of the MacMillan map [293].

Exercises

Exercise 8.1. Consider the perturbed MacMillan map (8.29) of Sect. 8.6.1.

(a) Prove that it is integrable for " D 0, by demonstrating that it possesses thepolynomial invariant I.xn; yn/ D x2

ny2n C x2

n C y2n � 2 xnyn. Investigate the

form of the invariant curves foliating the plane for different values of andshow that when j j > 1 the origin is a saddle point. In what way is the dynamicsfor > 1 different from < �1?

(b) Locate the stable and unstable manifolds of the linearized equations about theorigin, for D 1:6 and " D 0:001. Now follow them numerically in the full 2-dimensional plane starting with many initial conditions placed densely on thesemanifolds, very close to (0,0). Begin with ˇ D 0 and plot the intersections of themanifolds, delineating the “figure eight” region that you observe numerically atthe central part of the plane. Increasing by small steps the value of ˇ, can youlocate numerically the critical value ˇc beyond which the manifolds cease tointersect? Hint: This phenomenon of homoclinic tangency in two-dimensionalmappings is discussed in [151].


Problems

Problem 8.1. Consider the example of the Ikeda map [141]

�xnC1 D R C u.xn cos � yn sin /

ynC1 D u.xn sin C yn cos /(8.30)

where D C1 � C2=.1 C x2n C y2

n/, R, u, C1, C2 are free parameters. Its Jacobiandeterminant is J.R; u; / D u2, hence (8.30) is dissipative for juj < 1 and area-preserving for juj D 1.

(a) Fix the values of C1 D 0:4, C2 D 6 and R D 1 and show by plotting the iteratesof (8.30) in the .xn; yn/ plane that, for u D 0:9, all orbits converge on a strangeattractor. Now set u D 1 and demonstrate numerically the existence of a chaoticannular region surrounding a central domain about the origin, inside which themotion is predominantly quasiperiodic.

(b) Write a code implementing the procedure outlined in Sect. 8.6.1 and evaluatepdfs of the normalized variables s

.j /M for the parameter values u D 0:9, 1, for a

high number of iterations M and a large number of initial conditions Nic. Thus,show that the pdf of the strange attractor of the Ikeda map can be well fitted bya Gaussian.

(c) Consider now the area-preserving case u D 1, and select initial conditions inthe outer chaotic annulus surrounding the origin. Show that in this case the pdfsdo not converge to a Gaussian, but to a very different function, whose centralpart is well-fitted by a q-Gaussian with q D 5:3, even for very large M . Canyou detect any dynamical features on the boundary of this annular region thatmight justify its characterization as an “edge of chaos” regime?

Problem 8.2. It is possible to extend the study of Sect. 8.6 to higher dimensionalconservative maps and obtain results on their chaotic QSS and complex statistics.One such example is a 4D symplectic mapping model of accelerator dynamics [54,59] (see (5.21)). After some appropriate scaling, the equations of this map can bewritten as follows

�xnC1 D 2cxxn � xn�1 � �x2

n C y2n

ynC1 D 2cyyn � yn�1 C 2xnyn

(8.31)

where � D ˇxsx=ˇysy , cx;y � cos .2�qx;y/ and sx;y � sin .2�qx;y/, qx;y are theso-called betatron frequencies and ˇx;y are the betatron functions of the accelerator.Following [54], set qx D 0:21, qy D 0:24 and assume that ˇx;y are proportionalto q�1

x;y .

(a) Investigate weak diffusive phenomena in the y-direction of the four-dimensional phase space, as follows: Start by verifying first that for y1 D y0 D0 the point .x0; x1/ D .�0:0049; �0:5329/ is located within a thin chaotic layer


surrounding five large islands in the .xn; xn�1/ plane. Then, keeping the same.x0; x1/, choose .y1; y0/ very close to zero and observe the evolution of theyns, indicating the growth of the beam in the vertical direction as the numberof iterations M increases. Plot the projections of the iterates separately on the.xn; xnC1/ and .yn; ynC1/ planes.

(b) Write a code to evaluate the pdfs associated with these orbits, following theprocedure outlined in Sect. 8.6.1. Starting always with the same .x0; x1/, usedifferent initial values for y0 (y1 D 0) to compute pdfs of the normalized sumsof iterates of the yn-variable. Show that, just as in the case of two-dimensionalmaps, these distributions are initially of the q-Gaussian type, evolving intoexponential distributions and finally turning into Gaussians. Note, for example,that one such QSS with a maximum amplitude of about 0:00001 is apparent upto N D 219, when its amplitude is suddenly tripled in the y-direction.

(c) Demonstrate that the closer one starts to y0 D y1 D 0 the more the resultingpdf resembles a q-Gaussian, while the larger the y0; y1 values the faster the pdfstend towards a Gaussian-like shape.

Chapter 9Conclusions, Open Problems andFuture Outlook

Abstract The final chapter first summarizes and discusses the main conclusionsdescribed in the book. We then list a number of open problems, which we feel shouldbe further pursued in continuation of what we have presented in earlier chapters. Westart with some recent results that extend the mathematical theory of integrabilityfrom the viewpoint of singularity analysis and continue with some directions thatfurther develop the topics of nonlinear normal modes, localization, diffusion andthe complex statistical properties of nonlinear lattices. Finally, regarding the futureoutlook of research in Hamiltonian dynamics, we briefly review three topics of greatcurrent interest that were not treated in the book, but are extremely important inview of their far-reaching experimental applications: (1) anomalous heat conductionand the discovery of mechanisms that control heat flow based on the dynamics ofHamiltonian lattices, (2) soliton dynamics in nonlinear photonic structures and (3)kinetic theory of Hamiltonian systems with applications to plasma physics.

9.1 Conclusions

We hope that the preceding eight chapters have given the reader a fairly completepicture of what we believe are some of the most exciting and physically relevantresults of recent research in the field of Hamiltonian dynamics. Our purpose wasclearly not to dwell on the rigorous mathematical aspects of this research. What wewanted to do is present a number of fascinating developments, which demonstratethat multi-dimensional Hamiltonian systems still hold many secrets in their behaviorthat cannot be explained by some celebrated theorem and are not tractable byone of the classical approaches of local analysis or straightforward application ofperturbation theory.

Hamiltonian systems are not complex in the sense that one speaks of complexityin present day scientific literature. They are not described by complex networks,adaptive systems or agent based models and do not display self-organization, patternformation, or emergence of collective behavior. They are characterized, however, by


221

222 9 Conclusions, Open Problems and Future Outlook

varying degrees of order and chaos at different scales and exhibit fascinating out-of-equilibrium phenomena, long lived metastable states, emergence and breakdown oflocalized motion and surprising diffusion and transport properties. It is in this sensethat we have called Hamiltonian dynamics complex.

One important feature that characterizes the Hamiltonian systems treated in thisbook is their relevance to specific physical applications. The state of completeorder, expressed mathematically by the property of integrability (see Chap. 2) haspuzzled researchers since the times of Poincare and Kowalevskaya at the turn ofthe twentieth century. Despite the great activity inspired by the work of Painleve,from the early 1900s until today [98], very few examples have been found thatare completely ordered, possessing a full set of integrals. On the other hand, thecelebrated KAM theorem of the 1960s and Nekhoroshev’s work on the stabilityof near-integrable Hamiltonian systems [256] demonstrated that close to completeorder there is a vast number of physically important examples (the solar systembeing the most famous), whose behavior is globally ordered for “most” initialconditions in phase space and exponentially long intervals in time.

What happens then as one departs from integrability by varying the system’sparameters, or waits patiently to find out whether certain weakly chaotic orbits willreach an equilibrium state, where all fundamental modes share the same energy?The fundamental modes we speak of here are none other than the NNMs of Chaps. 3and 4, formed by the continuation of the linear normal modes of harmonic oscillatorsystems with nearest-neighbor interactions. NNMs play a very important role inthe study of local and global stability of Hamiltonian dynamics. They have beenespecially helpful in the investigation of the approach to “thermal” equilibrium inHamiltonian lattices ever since the discovery of FPU recurrences by Fermi, Pastaand Ulam in the mid 1950s [71], which remains to this day a topic of great researchinterest, as discussed in Chap. 6 of the book.

Chapter 5 discusses local dynamic indicators widely used in distinguishingordered from chaotic dynamics. More specifically, it reviews the topic of variationalequations and tangent dynamics of a reference orbit and focuses on a set ofparticularly efficient indices called SALI and GALI, which utilize more than onevariational vector to identify the nature of the orbit. Analytical results regarding thelong time asymptotic behavior of these indices were presented and a great numberof numerical experiments were described, verifying the validity of the asymptoticformulas and demonstrating their usefulness in multi-dimensional Hamiltoniansystems. Our main conclusion here is that the GALI method is indeed superior tomost other similar tools used in the literature for several reasons: First, it detectschaotic behavior faster than other methods, due to the exponential decay of theGALIks observed more rapidly in the plots of the higher k indices. It is also perhapsthe only method that can determine the dimensionality of a torus of quasiperiodicmotion and accurately predict its destabilization threshold.

One of the most important issues treated in this book to which the GALI methodapplies is the breakdown of FPU recurrences that constitutes the main theme ofthe results described in Chap. 6. In that chapter, we approach the problem ofFPU recurrences for the viewpoint of localization in Fourier (or modal q) space.

9.1 Conclusions 223

Using techniques of Poincare-Linstedt perturbation theory combined with amplenumerical evidence, we demonstrate that FPU recurrences can be understood interms of q-tori which reconcile the approach of q-breathers [128, 129] with of the“natural packet” scenario proposed in [37, 38]. Since these recurrences are relatedto the presence of stable q tori, the GALI method can be invoked to determineconditions under which q tori become unstable. This leads to a weakly diffusivemotion of nearby orbits and eventually to the collapse of recurrences and the onsetof energy equipartition in the FPU lattice.

These topics are closely related to the phenomena of localization and diffu-sion discussed in Chap. 7, this time in configuration (rather than Fourier) space.Indeed, the surprising occurrence of localized oscillations in nonlinear Hamiltonianlattices, pointed out as early as 1988 [306] and studied profusely in the literatureever since, has severe consequences for energy transport and diffusive processesarising in a multitude of applications, like Josephson junction networks, nonlinearoptical waveguide arrays, Bose-Einstein condensates and antiferromagnetic layeredstructures [125]. Furthermore, the breakdown of localization due to disorder andthe associated diffusive spread of wavepackets is at present a subject of intenseinvestigation [131, 317], in view of their many applications to light propagationin spatially random nonlinear optical media and related Bose-Einstein condensatephenomena.

Finally, we come to the remarkable complex statistical distributions of weaklychaotic motion in Hamiltonian systems discussed at length in Chap. 8. As withcomplete order, it turns out that complete disorder is also an exception in Hamil-tonian dynamics. Widespread global chaos, giving rise to Gaussian pdfs accordingto BG statistics, is known to prevail in the so-called Anosov or Sinai systems[11, 12, 307, 308] characterized by completely chaotic motion, such as elasticparticles in a box or a periodic array of scatterers, motion on surfaces of everywherenegative curvature, etc. To be sure, Gaussian pdfs and BG statistics were alsodiscovered by many researchers in large scale chaotic regions featured in mixedsystems, where domains of order and chaos coexist.

Complexity, however, in the sense discussed in the present book, is manifestedin domains of weak chaos, near the boundaries of ordered motion or in caseswhere stable periodic motion first destabilizes as the total energy is increased. Itis there that we find “thin layers” of chaotic orbits, whose pdfs in the sense ofthe Central Limit Theorem do not converge to Gaussians, even after very longtimes. It is in these cases where complex statistics arises associated with what wehave called metastable or quasistationary states in Chap. 8. Remarkably enough, ourinvestigations of these QSS share many common features with analogous studies ofdynamical systems characterized by long range interactions and strongly correlatedmotion [332].

The pdfs of sums of variables associated with this type of QSS are found tobe very well described by a class of functions called q-Gaussians related to thenotions of Tsallis’ entropy and nonextensive statistical mechanics, where the indexq satisfies 1 < q < 3 and q D 1 corresponds to the Gaussian distribution. A detailednumerical analysis of these pdfs demonstrates that their occurrence is far from


rare. Indeed, in many of the physically important Hamiltonian systems exploredin this way, we have discovered that these QSS are closely connected to energyequipartition in one-dimensional lattices, dynamical transitions in a microplasmaHamiltonian and other related phenomena [18]. Particularly with regard to thecharacterization of energy transport and diffusion in Hamiltonian lattices, it wouldbe of crucial importance to use the index q as a measure of how far our pdfs lie fromthe Gaussian equilibrium state of BG statistical mechanics.

9.2 Open Problems

As we all know, a book devoted to issues of active scientific interest cannot providethe final word on any of the topics treated in its chapters. Instead, what we have triedto do is communicate to the reader our strong conviction that a subject so classical,so thoroughly studied analytically, numerically and experimentally as Hamiltoniandynamics, still has many surprises to offer. The main reason for this, in our view,lies in the remarkable capacity of Hamiltonian dynamics to arise in many noveland exciting physical applications, where it continues to generate new directions ofresearch.

We would like to close this volume, therefore, by mentioning a number of openproblems, which we feel should be pursued, firstly because this will advance ourknowledge and expertise in the field of Hamiltonian systems. More importantly,however, progress in these issues will most likely provide answers to present dayquestions of great physical concern. Naturally, our point of departure will be thetopics we have presented in this book and hence the list of problems outlined belowis far from being comprehensive. Nevertheless, even in this limited context, webelieve that the reader will find our problems interesting enough to try them basedon the methods and information contained in the present volume.

9.2.1 Singularity Analysis: Where Mathematics Meets Physics

As we have seen in Chap. 2, the topic of integrability has been of great interest inthe study of Hamiltonian systems for centuries. One of the reasons, of course, is thatin this way one could study models, where the dynamics is perfectly ordered andunderstood. As the integrable examples, however, turned out to be a precious few,many physicists were discouraged and integrability gradually became a favouriteplayground for mathematicians. This situation was drastically changed in the late1960s, when soliton equations were discovered, which were not only physicallyinteresting [352] but could also be solved analytically by the method of InverseScattering Transforms (see e.g. [2, 3]). In this way, the abundance of applicationsof soliton equations in water waves, nonlinear optics, Josephson junctions and evenfield theory has kept the subject of integrable PDEs alive for the past 40 years.

9.2 Open Problems 225

Such was not the case, however, with integrable systems of ODEs describingHamiltonian dynamics. After the celebrated discovery of the Toda lattice [329]and the Ablowitz-Ladik model [4], the search for integrable Hamiltonian systemsdid not produce many physically interesting examples [60]. Despite a thoroughexploration using the tools of Painleve analysis [98, 279] the harvest of newintegrable Hamiltonians has not been particularly satisfying.

Nevertheless, an interesting conclusion did arise concerning the case of near-integrable Hamiltonians, where the Painleve requirement allowing only poles asmovable singularities was lifted and multi-valued solutions emerged in the complexdomain. Ziglin’s theory of non-integrability [152] showed that small perturbationsof a Painleve integrable system produced infinitely branched solutions, typicallycontaining logarithmic singularities. More importantly, however, the coefficients ofthe logarithmic terms in the singular expansions were shown to be related to theMel’nikov integral [160], which is known to provide an estimate of the width of thechaotic layer in nearly-integrable systems [152, 287, 288].

The next question, of course, is what happens when the solutions of a Hamil-tonian system are locally finitely branched. As was demonstrated in a number ofpapers [51, 113, 132], local finite branching is not sufficient to ensure integrability.It is important that the global Riemann surface be finitely sheeted. What oftenhappens in non-integrable systems is that, as one repeatedly integrates along a closedloop in the complex time plane, new singularities appear, which preclude the returnto the initial values and produce an infinitely sheeted solution surface, even thougheach singularity is by itself finitely branched. Every time a system of ODEs withfinitely branched singularities turned out to also possess a finitely sheeted Riemannsurface, a known integrable system was recovered which can be transformed to onethat has the full Painleve property after appropriate variable transformations [152].

Thus, an open problem is to discover a new integrable ODE (or systemsof ODEs), whose solutions are locally finitely branched with a finitely sheetedRiemann surface, but which has not yet been identified by the Painleve analysis.An attempt in that direction was made in [132] in the framework of perturbationtheory. Unfortunately, in all the cases tried, the finitely sheeted property establishedat first order was not preserved at higher orders.

A related problem is to connect the structure of the finitely sheeted Riemannsurface of the solutions of a problem with finetely branched singularities to thedynamical properties of the orbits. An interesting start in this direction was made in[69,70], where an integrable model of this type was analysed and certain remarkableproperties of its solutions (such as the isochrony of its periodic orbits) were found tobe related quantitatively to the structure of its Riemann surface. A next step mightbe to vary slightly the parameters of the model and investigate the onset of chaoticmotions in connection with changes in the geometry of the Riemann surface.

Another important problem is to use the analysis described in Chap. 7 of [152]and compute the Mel’nikov vector of a nearly integrable Hamiltonian systemusing information about the singularities of its solutions on the invariant manifoldsassociated with its saddle points. The usefulness of this approach lies in the fact thatone does not need to know explicitly the corresponding homoclinic (or heteroclinic)


solutions of the integrable system to perform this computation. Estimating themagnitude of the components of the Mel’nikov vector, the challenging question isto extract some information about the width of the chaotic layer of the system underdifferent projections of its dynamics in the multi-dimensional phase space.

9.2.2 Nonlinear Normal Modes, Quasiperiodicityand Localization

Let us recall that in Chap. 3 the problem of stability of motion was approachedstarting from certain very simple periodic solutions of N dof Hamiltonians likethe FPU�ˇ one-dimensional lattice, under periodic and fixed boundary conditions.One reason for proceeding in this way is the fact that these periodic solutions areexpressed in terms of known functions and the study of their (linear) stability leadsto the analysis of a single second order ODE of the Lame type.

Concerning in particular the solutions called SPO1 (3.26) and SPO2 (3.27) of theFPU-ˇ model under fixed boundary conditions, we observed in Sect. 3.2 that SPO1destabilizes at much higher energies than SPO2. In fact, the destabilization thresholdof SPO2 as a function of the number of particles N was seen to follow the law (3.39),Ec _ N �2, derived in [128] for the low q D 1; 2; 3; :: modes, where q numbers theNNM continuations of the linear modes of the system. And yet our SPO2 solutioncorresponds to a much higher mode with q D 2.N C 1/=3! Why is that so? Howcan the instability threshold of such a high mode—even in its asymptotic form—coincide with the one satisfied by the low q modes? We do not know.

The SPO1 solution on the other hand, is identified with the q D .N C1/=2 modeand satisfies a very different asymptotic destabilization law of the form Ec _ N �1.The open question, therefore, is: What is the theory that governs the stability of thehigher q NNMs as N ! 1? We do not speak here of course of the highest q modesat the right end of the spectrum (q D : : : N � 2; N � 1; N ), for which the analysisis very similar to that of the lowest ones, at least for the FPU-ˇ chain under fixedboundary conditions.

It is well-known, after all, that most studies carried out to date on the stabilityof NNMs in these FPU lattices concern the low q modes, since they are the onesthat play the main role in the phenomenon of FPU recurrences (see Chap. 6).What then is the significance of the higher modes? Are they responsible for someother physically important phenomenon? Does their instability threshold obey anasymptotic law different than those we have mentioned so far and what is the deepermeaning of these laws? Again we don’t know.

One way to approach this problem, at least for the case of periodic boundaryconditions is provided by the theory and results of Chap. 4 (see e.g. Sect. 4.4.2). Inthat theory, symmetry properties of the equations of motion are exploited to formbushes of orbits consisting of linear combinations of fundamental NNMs. Studyingthen the interactions among the modes belonging to each bush, it is possible to


study analytically the linear stability of the bush itself, which generally representsa quasiperiodic solution of the system whose frequencies are those of the modesparticipating in the bush. Thus, it becomes possible to explore stability properties ofthese Hamiltonians more globally, as has been done e.g. in [82, 87] (see a review ofthis approach in [64]).

An open problem, therefore, is to combine the results of the bush theory withthe study of stability of low-dimensional tori (called q-tori) discussed in Chap. 6 toinvestigate more deeply the phenomenon of FPU recurrences. This effort may meetwith limited success for lattices with fixed boundaries, due to the small numberof symmetries present in that case. Our suggestion, therefore, is to study the FPUrecurrence problem for periodic boundary conditions from the point of view of bushtheory and compare the results with the q-tori approach of Chap. 6. Note that caremust be taken to deal with the twofold degeneracy of the linear spectrum in periodicmodels, which modifies somewhat the analysis of the recurrence problem [272].

Let us now come to the question of stability of tori of quasiperiodic motion and itsconnection with the phenomenon of localization in nonlinear lattices. As the readerrecalls, in Chap. 6 we interpreted the phenomenon of FPU recurrences as a resultof localization in the Fourier modal space and studied in Sect. 6.3.1 the stability ofthe associated q-tori using the GALI methods of Chap. 5. What about localizationin configuration space as exemplified e.g. by the occurrence of discrete breathers,like those investigated in Chap. 7? Do we also find low-dimensional tori in theirneighborhood?

In [63] this question was answered affirmatively in the case of a lattice modeldescribed by the Hamiltonian

H.t/ DNX

nD1

�1

2Px2n C 1

2Œ1 � � cos.!d t/�x2

n � 1

4x4

n C K

4.xnC1 � xn/4

�; (9.1)

in which harmonic nearest-neighbor interactions are absent and a periodic forcingterm has been added with amplitude � and frequency !d . This Hamiltonian wasstudied in detail in [241], where among other phenomena chaotic breathers werealso observed that remain localized on a small number of sites for very long times.As was explicitly shown in [63] via the GALI criterion, in the neighborhood ofcertain fundamental discrete breathers of (9.1), low-dimensional tori are seen toexist, which become unstable as the parameters of the problem are varied.

The following question arises therefore: Do these localization phenomenadepend on the fact that (9.1) is free from linear modes that can destroy localizedoscillations through resonances with the phonon spectrum? It would be veryinteresting to investigate this question by adding to the above Hamiltonian aharmonic part of the form

PNnD1f L

2.xnC1 � xn/2g, and slowly vary L � 0 to

determine how these terms affect the behavior of the dynamics.As we also found out in Chaps. 5 and 8, another class of models that one can

study in connection with Hamiltonian dynamics are symplectic maps, which arenumerically far easier to explore and arise in many physical applications [42,50,58,


180, 244, 269]. Thus, it might be quite illuminating to represent each oscillator by atwo-dimensional symplectic map (e.g. of the standard map type [136,214,246]) andconsider a one-dimensional “lattice” formed by N such maps coupled by nearestneighbor interactions as follows

xjnC1 D xj

n C yjnC1;

yjnC1 D yj

n C Kj

2�sin.2�xj

n / � ˇ

2�fsinŒ2�.xj C1

n � xjn /� C sinŒ2�.xj �1

n � xjn /�g(9.2)

where j D 1; : : : ; N , ˇ is the coupling parameter between the maps and fixedboundary conditions x0 D xN C1 D 0 are assumed.

This model was investigated in [63], where it was demonstrated numerically thatit does possess discrete breather solutions, around which low-dimensional tori canbe identified, as long as the discrete breather is stable. Furthermore, changing tolinear normal mode coordinates, in which the linear parts of the maps are uncoupled,it is easy to show that when the energy is placed initially in s D 1; 2; 3; : : : ; ofthese modes, and the parameter ˇ is small enough, FPU-like recurrences arise,accompanied by the presence of s-dimensional tori, just as we found in the FPUHamiltonian models. Furthermore, the stability of these tori and the breakdownof recurrences as ˇ increases can also be accurately monitored using the GALImethod [63].

It would, therefore, be very interesting to extend these studies and investigate ingreater detail dynamical phenomena in coupled standard maps, rather than pursuingthem on Hamiltonian lattices, which is computationally far more time-consuming.For example, it would be very interesting to impose periodic boundary conditionsto the coupled map system (9.2) and try to understand its bushes of orbits, FPU typerecurrences and their effect on the onset of energy equipartition in such models.

Finally, it is important to mention certain open questions related with the stabilityof discrete breathers in Hamiltonian lattices. For instance, why is it that simplebreathers (exhibiting only one exponentially localized central maximum) are sofrequently found to be stable under small perturbations? More specifically, let usconsider multibreathers, which represent localized oscillations with more than onemajor maximum (or minimum) and compute all of them for a one-dimensionallattice of N particles with harmonic nearest neighbor interactions and quartic on-site potential, using homoclinic orbit methods [39, 42]. Let us also associate themsymbolically with sequences of 0; C; �, corresponding to sites of small (or nooscillation), large positive and large negative amplitude respectively [41]. Whyis it that most of them are found to be unstable under small changes of theirinitial conditions [40]? More remarkably, how can one explain the observation thatthis instability characterizes all multibreathers which contain one (or more) of thecombinations C�, C0C, or C00C in their symbolic representation?


9.2.3 Diffusion, Quasi-stationary States and Complex Statistics

And now we come to one of the most engaging aspects of complex Hamiltoniandynamics, which concerns the phenomenon of diffusion, by which we mean slowand gradual transport of energy in weakly chaotic domains. This transport concernseither modes in Fourier space, as we found out while studying the breakdown ofFPU recurrences in Chap. 6 (see Sect. 6.4), or particle motion in configuration space,resulting from the breakdown of localized disturbances in disordered lattices, as wediscovered in Sect. 7.4.2.

Let us start with diffusion in modal space and recall the results presented inSect. 6.4.1 on what we called stage II of the dynamics of the FPU-˛ lattice. We hadshown first that the model exhibits a slow transport of its total energy E from aninitial “packet” of low modes to the tail modes, on its way to energy equipartition asE is increased. Studying the rate of this process, we observed that it is sub-diffusive(_ t� , with � < 1) and had attempted to estimate the time it takes for the systemto reach equipartition, using the theoretical prediction of the diffusion rate of themotion described in Sect. 6.4.2.

More specifically, we defined a quantity �.t/ (6.44) as the sum of normalizedharmonic energies of the upper third of the mode spectrum and plotted in Fig. 6.10�.t/ as a function of t , for E D 0:01, E D 0:1 and E D 1. We thus observed adiffusion law of the form � / Dt� , demonstrating first that the process indeedis one of subdiffusion. Attempting then to estimate the time interval T eq requiredfor energy equipartition in Sect. 6.4.3, we found T eq � E�3, in agreement withan analogous formula given in [112], for energies E 2 Œ0:4; 2�. This prediction,however, disagrees with the results of numerical integration for 0:1 < E < 0:4,while for E � 0:1, it is impossible to check its validity by direct computation.Indeed, for E D 0:01 the value of the exponent � is practically zero.

The open problem, therefore, is to extend the studies we have described for theFPU-˛ model to energy regimes where all attempts of the kind mentioned abovehave failed. To date we do not know of a theoretical estimate that allows us toreliably predict, even for one-dimensional lattices of the FPU type, the time T eq

required for equipartition when E becomes too small. Thus, we can only guess:Either equipartition times become exponentially long, or the system remains trappedon the neighborhood of a q-torus for all time and equipartition never occurs belowa certain energy threshold.

This brings us to a similar question regarding diffusion in one-dimensionalHamiltonian lattices in the presence of disorder, discussed in Sect. 7.4 of Chap. 7:What happens if nonlinearity is introduced to the system? How can we understandits effect on the localization properties of wave packets in disordered systems? Thischallenging problem has been investigated in recent years by many researchers[47, 48, 124, 131, 144, 202, 210, 252–255, 271, 311, 317, 338, 339]. Most of thesepapers consider the evolution of an initially localized wave packet and show thatwave packets spread subdiffusively for sufficiently strong nonlinearities.


On the other hand, if the nonlinearities are weak enough, wave packets appearto be frozen, in the sense of Anderson localization, at least for finite integrationtimes. In fact, an alternative interpretation regarding wave packet evolution has beenconjectured in [178] claiming that these states may be localized on some KAMtorus for infinite times! Which of these interpretations is correct? Do wave packetscontinue to spread subdiffusively in some weakly chaotic domain for all time, or dothey eventually settle down on a finite (or infinite) dimensional torus?

We suggest here that this question may also be studied by a statistical approach,similar to the one we have described in Chap. 8 of this book. As we have explainedin that chapter, chaotic orbits in weakly chaotic domains of multi-dimensionalHamiltonian systems are often seen to form QSS that are extremely long lived. Thisis exemplified by pdfs of sums of their variables, which, in the sense of the CLT, donot converge to Gaussians, but are well-described by a class of so-called q-Gaussianfunctions, whose index lies in the interval 1 < q < 3 and tends to q D 1 when theQSS reaches the Gaussian state of strong chaos.

An interesting open problem, therefore, may be formulated as follows: Considerthe QSS formed by the spreading of a wave packet resulting from the delocalizationof a central excitation in a disordered KG or DNLS one-dimensional lattice.Evaluate the sum of the position coordinates of all particles and study the timeevolution of its normalized pdf as it evolves in a presumably weakly chaotic regionof phase space. In the event that this pdf is well approximated by a q-Gaussian,estimate the index q and plot its behavior as a function of time. If its values show atendency to approach q D 1 as time progresses then the process can be characterizedas chaotically diffusing, while if the q values remain far from unity for all time, thiswould constitute evidence that the dynamics eventually “sticks” to the boundary ofsome high-dimensional torus of the system.

Finally, we mention some additional open problems that concern the complexstatistics of weakly chaotic motion in Hamiltonian systems. In Chap. 8, we saw thatit was possible to find QSS that can be fitted well by q-Gaussian pdfs for a number ofHamiltonian systems mainly of the FPU type [18]. This has been primarily achievedin the neighborhood of certain NNMs of the FPU-ˇ model, at energies where thesemodes first turn unstable as the total energy E is increased. Examples of such NNMswere the �-mode in the case of periodic boundary conditions and the SPO1 andSPO2 modes for fixed boundary conditions [18].

The obvious question then is: Where else in the multi-dimensional phase spacecan one find weakly chaotic regimes, where similar QSS phenomena are observed?Are these important enough physically to justify the use of nonextensive statisticalmechanics to characterize the dynamics in the corresponding regimes? One suchpossibility already mentioned above concerns the spreading of wave packets indisordered nonlinear lattices. Another one refers to the role of q-Gaussian approx-imations of pdfs for chaotic orbits associated with the breakdown of recurrences inthe FPU-˛ model, which was recently studied in [15]. Are there any other examplesof Hamiltonian dynamics, where this type of complex statistics might be relevant?

9.3 Future Outlook 231

9.3 Future Outlook

9.3.1 Anomalous Heat Conduction and Control of Heat Flow

The problem of heat conduction in dielectric crystals has been an important issue inmathematical physics for many years. The first celebrated approach to this problemis due to none other than Rudolf Peierls (1907–1905), who proposed in the 1930s anexplanation based on the fundamental assumption that the nonlinearity of phononinteractions can lead to a state of thermal equilibrium, at which normal diffusioncan occur with constant heat conductivity [264].

Ever since the famous FPU experiments of the 1950s, however, there has been animpressive amount of analytical and numerical work by many authors, who showedon a wide variety of lattice models that nonlinearity is not sufficient to ensure energyequipartition between the phonon modes and normal thermal conduction. Indeed,even in cases where normal heat conductivity was observed, the heat conductioncoefficient often depends strongly on the model and differs significantly from whatis measured in actual physical experiments [219].

As noted by Casati and Li in [74], one-dimensional FPU lattices exhibitanomalous heat conduction, whose thermal conductivity coefficient � is found todiverge with the system size L as � � L2=5, while when transverse motions areincluded in the model one finds � � L1=3 [343]. In fact, a simple formula has beenproposed relating heat conductivity with anomalous diffusion in a one dimensionalsystem [221], as follows: Let us recall that Fourier’s law of heat conduction statesthat the heat flux j is proportional to the temperature gradient rT as

j D ��rT; (9.3)

this implies that j is independent of the size of the system. Denoting energydiffusion by < �2 >D 2Dt˛; .0 < ˛ 5 2/, it has been demonstrated in [221]that � / Lˇ , with ˇ D 2 � 2=˛!

This latter relation, corresponding to subdiffusion for ˛ < 1 and superdiffusionfor ˛ > 1, was found to be in good agreement with many results available from one-dimensional lattices, as well as existing data from many physical systems [74]. Onthe other hand, the situation regarding the divergence of the conductivity coefficientwith system size in two and three spatial dimensional FPU models is still a topic ofongoing investigation [220, 302].

Another major development in this field has been the realization that exponentialdynamical instability is a sufficient but not necessary condition for the validity ofFourier’s law (9.3) [9, 76, 222–224]. Indeed, in a model of a Lorentz gas “channel”,represented by two parallel lines and a series of semicircles of radius R placed alongthe channel in a “triangular” way so that no particle can move in the horizontaldirection without colliding with the disks, accurate numerical evidence has shown[9] that heat conduction does obey Fourier’s law. As the dynamics of the Lorentzgas is rigorously known to be mixing with positive Kolmogorov-Sinai entropy,


this result clearly suggests that mixing with positive Lyapunov exponents is a“sufficient” condition for the validity of Fourier’s law of heat conduction.

On the other hand, when the semi-circles in the above model are replaced bytriangles, with all angles irrational multiples of � , it has been shown [75] that theLyapunov exponent is zero and yet the model exhibits unbounded Gaussian diffusivebehavior. This strongly suggests that normal heat conduction can take place evenwithout the strong requirement of exponential instability.

In another series of studies, researchers considered the possibility of actuallybeing able to control the heat flow, in a variety of one-dimensional nonlinear lattices.Based on a model proposed in [326], in which heat was numerically shown to flowin one preferential direction, Li, Wang and Casati [225] considered an improvedversion described by the Hamiltonian

H DNX

iD1

�p2

i

2mC 1

2k.xi � xiC1 � a/2 � V

.2�/2cos.2�xi /

�; (9.4)

and divided the corresponding lattice in three parts: On the left part (L) the particlesare assigned a spring constant k D kL and on site parameter V D VL, while onthe right part (R) k D kR D kL and V D VR D VL. The two ends of the latticeare connected to heat reservoirs at temperatures T D TL D T0.1 C /, T DTR D T0.1 � / and in the middle part of the lattice the spring constant was setto k D kint . Fixing now the values of a D m D 1, VL D 5, kL D 1, one mayadjust the parameters ; ; kint ; T0 and plot the heat current as a function of fordifferent values of T0.

As is well-known, (9.4) is the Hamiltonian of the Frenkel-Kontorova model,which has been shown to exhibit normal heat conduction [172] in the homogeneouscase kL D kint D kR D k; VL D VR D V . However, setting kint D 0:05; D 0:2

and N D 100 a remarkable phenomenon was observed: For > 0 the heat currentjC was found to increase with , while in the region < 0 the heat current j�is almost zero, implying that the system behaves as a thermal insulator! In fact, byincreasing the value of T0 D 0:05; 0:07; 0:09 the rectifying efficiency, defined asjjC=j�j can increase by as much as a few hundred times [225]!

Thus the study of these models has opened up the way for a multitude ofapplications of great importance such the designing of novel thermal materialsand devices like thermal rectifiers [225, 326] and thermal transistors [230]. Indeed,the possibility of building such devices has been demonstrated in many laboratoryexperiments [79,192,270,348], whose discoveries may eventually lead to practicallyuseful products.

9.3.2 Complex Soliton Dynamics in NonlinearPhotonic Structures

The behavior of light in media with nonlinear optical response is one more topic ofintense research interest, where Hamiltonian dynamics is involved. The main reason


for this is the fact that nonlinear effects in light propagation such as self-localizationand nonlinear interactions have far-reaching technological applications in opticaltelecommunications, medicine and biotechnology. The invention of the laser alongwith recent developments in material science and the advent of novel nonlinearoptical materials have led to an explosion of experimental and theoretical studiesin nonlinear optics. More specifically, nonlinear effects in optical devices and thecomplexity of the associated wave dynamics have been recognized as effectivepotential mechanisms for light control.

One of the most fundamental phenomena in nonlinear optical wave propagationis the self-trapping of spatially localized beams, resulting from the balance betweendiffraction and nonlinearity. The concept of the soliton, as a solitary wave whichremains intact under mutual interactions (see Sect. 6.1.1), has played a crucial role inthe development of nonlinear optics [190,330]. Solitons are obtained as solutions ofPDEs corresponding to infinite-dimensional integrable Hamiltonian systems and areremarkably robust under small perturbations [2,3]. In the context of nonlinear optics,wave propagation is accurately modeled by the Nonlinear Schrodinger Equation(NLS) under the so-called paraxial approximation [215].

Integrability, however, as we have so often said, is the exception. Once spa-tial inhomogeneities are allowed in the system the equations describing wavepropagation become non-integrable due to symmetry breaking. This excludes theexistence of pure solitons in the strict mathematical sense, but gives rise to theoccurrence of a plethora of solitary waves, which do not have a counterpart inthe homogeneous case [91, 182, 183, 216]. More importantly, it results in complexand interesting wave dynamics, allowing for the control and routing of lightbeams. For example, an unstable stationary solution could evolve into a stableone under certain circumstances, providing a transition phenomenon with manypotential applications. Furthermore, many studies have shown that different types ofinhomogeneity can be very useful for designing photonic structures having desirablefeatures and properties. The latter are expected to provide optical devices withan all-optical circuitry capable of switching, routing, information processing andcomputing beyond the limitations of electronics-based systems. It is also importantto emphasize that all the above concepts and effects are relevant to many otherdomains of nonlinear physics such as the formation of spin waves in magnetic filmsand the rapidly developing field of coherent matter waves and nonlinear atom optics[65, 260].

Now, wave propagation of the electric field variable u D u.z; x/ in an inhomoge-neous medium with Kerr-type nonlinearity is described by the perturbed NLS

i@u

@zC @2u

@x2C 2juj2u C "

�n0.x; z/ C n2.x; z/juj2� u D 0 (9.5)

where z and x are the normalized propagation distance and transverse coordinaterespectively and the potential functions ni .x; z/; i D 0; 2 model the spatialvariation of the linear and nonlinear refractive indices. The dimensionless parameter" indicates the strength of the potential and is “small” for relatively weakly


inhomogeneous media, which is the case of interest in technological applications.The functions ni .x; z/; i D 0; 2 can be either periodic or nonperiodic in x and z.

As is well-known, the unperturbed NLS ((9.5) with " D 0), has a fundamentalsoliton solution of the form

u.x; z/ D AsechŒA.x � x0/�ei. v

2 xC2�/ (9.6)

where x0 D x0.z/ D vz, � D �.z/ D �zv2=8 C zA2=2, A is the amplitude orthe inverse width of the soliton solution, x0 is its center (called “center of mass”due to an analogy between solitons and particles), v the constant velocity, and � thenonlinear phase shift.

The longitudinal evolution of the center x0 under the lattice perturbation isobtained by applying the effective-particle method [191]. According to this methodone assumes that the functional form as well as other properties of the soliton (width,power) are conserved in the case of the weakly perturbed NLS. This assumptionhas to be verified, however, through numerical integration of (9.5) at least to agood approximation. The motion of the center x0 is equivalent to the motion ofa particle with mass m � R juj2dx D 2A under the influence of an effectivepotential Veff .x0; z/ D 2 �

R �n0.x; z/ C n2.x; z/ju.xI x0/j2� ju.xI x0/j2dx and

is described by the nonautonomous Hamiltonian

H D mv2

2C Veff .x0; z/ (9.7)

where z is considered as “time” and v D Px0 (the dot denotes differentiation withrespect to z) is the velocity of the soliton’s “center of mass”.

In the case of zero longitudinal modulation (@Veff =@z D 0) the system isintegrable and soliton dynamics is completely determined by the form of thez-independent effective potential. Stable and unstable solitons can be formed atthe minima and maxima of the effective potential, while solitons can also beeither trapped between two maxima of the effective potential or reflected bypotential barriers. Soliton dynamics in several photonic structures can be describedunder this approach. For the simpler case of harmonic (sinusoidal) transversemodulation of the linear refractive index, the positions of the stable and unstablesolitons are independent of the soliton “mass” (power) [119]. For more complexmodulations of the linear and nonlinear refractive indices, as well as modulationswith incommensurate wavelengths, soliton positions and stability depend stronglyon the specific soliton power [196]. The latter results in additional functionalityproviding power-controlled discrimination of such structures. In addition, semi-periodic photonic structures can also be described such as interfaces between twoperiodic lattices with different characteristics as well as interfaces between periodiclattices and homogeneous media. Thus, it is shown that power-dependent solitonreflection, transmission, and trapping can be achieved by the formation of effectivepotential barriers and wells [197].

The presence of an explicit z dependence in the effective potential (@Veff =@z ¤ 0)results in the nonintegrability of the Hamiltonian system which describes the


soliton motion and allows for a plethora of qualitatively different soliton evolutionscenarios. The corresponding richness and complexity of soliton dynamics opensa large range of possibilities for interesting applications where the underlyinginhomogeneity results in advanced functionality of the medium. Nonintegrabilityresults in the destruction of the heteroclinic orbit (separating trapped from travellingsolitons), allowing for dynamical trapping and detrapping of solitons. Therefore,we can have conditions for enhanced soliton mobility: Solitons with small initialenergy can travel through the lattice as well as hop between adjacent potential wellsand dynamically be trapped in a much wider area. Moreover, resonances betweenthe frequencies of the internal unperturbed soliton motion and the z-modulationfrequencies result in quasiperiodic trapping and symmetry breaking with respect tothe velocity sign. It is worth mentioning that all these properties depend strongly onsoliton characteristics, such as its amplitude A (see (9.6)), so that different solitonsundergo qualitatively and quantitatively distinct dynamical evolution in the sameinhomogeneous medium [263].

For strong modulations of the optical medium properties the effective particleapproach is no longer valid. However, there are several cases where analyticalsolutions can still be found such as the case of a nonlinear Kronig-Penney modelconsisting of interlaced linear and nonlinear regions. The underlying systems arewithin a large class of piecewise linear systems for which analytical solutions canbe constructed by utilizing the phase space of the system [194, 200]. Followingthis approach, solitary wave solutions have been obtained for periodic [193, 195]and nonperiodic [198] structures, which can be further utilized in perturbativetechniques for the exploration of new classes of solutions.

Thus, the study of soliton dynamics in complex photonic structures is a chal-lenging research direction, as it promises to provide light control functionality inappropriately designed media with numerous technological applications. It is atypical case of a field where complexity opens new possibilities with significanttechnological advantage, and where the concepts and methods of Hamiltoniandynamics provide excellent tools for understanding and designing practical systemswith desirable properties.

9.3.3 Kinetic Theory of Hamiltonian Systems and Applicationsto Plasma Physics

The phase space evolution of particle distribution functions of non-integrableHamiltonian systems is usually described by a quasilinear theory leading to anaction diffusion equation, in which the perturbation terms are taken into accountthrough a diffusion operator D that generally depends on both space and time[114, 337]. This diffusion equation may be derived starting with the Fokker Planckequation

@F

@tD � @

@J.VF / C 1

2

@2

@2J.DF /; (9.8)


where J is the action variable in x; y; z space. Noting now that [201, 226]

V D 1

2

@D@J

; (9.9)

the desired diffusion equation is directly obtained as

@F

@tD @

@J

�D

@F

@J

�: (9.10)

In the standard derivations of the above equations it is assumed that motion israndomized, with respect to the phase of the perturbation, after one period. Thisis related to the Markovian assumption, used for example in studying Brownianmotion, according to which the dynamics is characterized by completely uncorre-lated particle orbits, phase-mixing, loss of memory and ergodicity. These statisticalproperties naturally lead to the important simplification that the long time behaviorof particle dynamics remains the same after one interaction time with the wave. Thesignificant drawback is that the diffusion coefficient is singular, with a Dirac deltafunction singularity so that further considerations are necessary in order to utilizesuch operator in theoretical and numerical studies [114, 337].

Unfortunately, the Markovian assumption runs contrary to the dynamical behav-ior of particles interacting with coherent perturbations, since phase space is oftena mixture of chaos and order with islands of coherent motion embedded withinchaotic regimes [226]. Furthermore, near the boundaries of such islands particlesare observed to “stick” and undergo coherent motion for times much longer thanthe interaction time. Even when the amplitude of the perturbations is assumed to beimpractically large so that the entire phase is chaotic, the quasilinear theory fails togive an appropriate description of the evolution of the distribution function [280].The persistence of long time correlations invalidates the Markovian assumption[36, 73, 301, 353].

Therefore, we need to develop a kinetic theory, based on first principles, in orderto study a large variety of realistic systems for which the Markovian assumptionsare simply not valid. Recently, such a theory was proposed using Hamiltonianperturbation theory and Lie transform techniques to derive a hierarchy of kineticevolution equations that does not rely on any simplifying statistical assumptions[201]. The final kinetic equation in the hierarchy is obtained by averaging over allangles and represents an evolution equation for the distribution function in actionspace with an operator that is nonsingular and time-dependent. Moreover, thisequation is time-reversible and is capable of describing the transient time evolutionof the action distribution function as well as a range of dynamical phenomena muchricher and wider than simple diffusion.

Let us consider the Hamiltonian corresponding to a generic system

H.J; �; t/ D H0.J/ C �H1.J; �; t/: (9.11)


The Hamiltonian consists of two parts: the integrable part H0.J/ that is a function ofthe constants of the motion J of a particle moving in a prescribed equilibrium field,and

H1.J; �; t/ DX

m¤0

Am.J/ei.m��!0

mt / (9.12)

which includes perturbations to the equilibrium field. � are the angles canonicallyconjugate to the actions J and t is time. The complex frequency !0

m D !m C i�m

allows for steady state (�m D 0), growing (�m > 0), or damped (�m < 0) waves.Note that for the case of a charged particle in a toroidal plasma, in the guiding centerapproximation for an axisymmetric equilibrium [186], the three actions are themagnetic moment, the canonical angular momentum, and the toroidal flux enclosedby a drift surface. The respective conjugate angles are the gyrophase, azimuthalangle, and poloidal angle. The perturbation terms correspond to electromagneticwaves and � is an ordering parameter indicating that the effect of H1 is perturbative.

According to the proposed kinetic theory [201], the operator D in (9.10) isgiven by

D.J; t/ DX

m¤0

mm jAm.J/j2 e2�mt

˝m.J/2 C �2m

n�m

�1 � e��mt cos .˝m.J/t/

� C

C˝m.J/e��mt sin .˝m.J/t/o

(9.13)

where ˝m.J/ D m � !0.J/ � !m and mm is a dyadic.In the limit t ! 1 and �mt ! 0, (9.13) leads to the standard time-independent

quasilinear diffusion tensor

Dql.J/ DX

m¤0

mm jAm.J/j2 ı .˝m.J// ; (9.14)

where ı is Dirac’s delta function. The long time limit is justified only for statisticallyrandom, or Markovian processes [226], while the delta function excludes shorttime transient effects and is difficult to implement numerically. More importantly,the asymptotic time limit results in a time-irreversible Fokker Planck equation,while the time-dependent D in (9.13), being an odd function of time, leads to atime-reversible evolution equation. In contrast to the traditional quasilinear theory[114, 337], the kinetic evolution equation (9.10) with the time-dependent operator(9.13) does not distinguish between resonant and nonresonant particles and includesboth growing and damped waves. In the vicinity of resonances given by ˝m D 0, Dis continuous and non-singular even when �m D 0 and the width of the resonancedecreases with time.

This kinetic theory is expected to provide an accurate description of manycomplex realistic systems of technological interest. Among the most importantones are fusion plasmas, where charged particles are confined by a static magnetic


field and interact with either static magnetic perturbations or perturbations byradio frequency waves injected to heat the plasma, drive the current, and suppresslocal instabilities [186, 199]. In addition, the theory is directly applicable to spaceplasmas, particle acceleration studies and, in general, any Hamiltonian systemwhere chaotic motion coexists with regular motion in a strongly inhomogeneousphase space.

References

1. F. Abdullaev, O. Bang, M.P. Sørensen (eds.), Nonlinearity and Disorder: Theory and Appli-cations. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 45 (Springer,Heidelberg, 2002)

2. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scat-tering. London Mathematical Society Lecture Note Series, vol. 149 (Cambridge UniversityPress, Cambridge, 1991)

3. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia,1981)

4. M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear SchrodingerSystems (Cambridge University Press, Cambridge, 2004)

5. E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Scaling theory oflocalization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676(1979)

6. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)7. O. Afsar, U. Tirnakli, Probability densities for the sums of iterates of the sine-circle map in

the vicinity of the quasiperiodic edge of chaos. Phys. Rev. E 82, 046210 (2010)8. Y. Aizawa, Symbolic dynamics approach to the two-dimensional chaos in area-preserving

maps. Prog. Theor. Phys. 71, 1419–1421 (1984)9. D. Alonso, R. Artuso, G. Casati, I. Guarneri, Heat conductivity and dynamical instability.

Phys. Rev. Lett. 82, 1859–1862 (1999)10. P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505

(1958)11. D.V. Anosov, Geodesic flows on a compact Riemann manifold of negative curvature. Trudy

Mat. Inst. Steklov 90, 3–210 (1967). English translation, Proc. Steklov Math. Inst. 90, 3–210(1967)

12. D.V. Anosov, Y.G. Sinai, Some smooth Ergodic systems. Russ. Math. Surv. 22(5), 103–167(1967)

13. Ch. Antonopoulos, T. Bountis, Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice. Phys. Rev. E 73, 056206 (2006)

14. Ch. Antonopoulos, T. Bountis, Detecting order and chaos by the linear dependence index(LDI) method. ROMAI J. 2, 1–13 (2006)

15. Ch. Antonopoulos, H. Christodoulidi, Weak chaos detection in the Fermi-Pasta-Ulam-˛system using q-Gaussian statistics. Int. J. Bifurc. Chaos 21, 2285–2296 (2011)

16. Ch. Antonopoulos, T.C. Bountis, Ch. Skokos, Chaotic dynamics of N-degree of freedomHamiltonian systems. Int. J. Bifurc. Chaos 16, 1777–1793 (2006)

T. Bountis and H. Skokos, Complex Hamiltonian Dynamics,Springer Series in Synergetics, DOI 10.1007/978-3-642-27305-6,© Springer-Verlag Berlin Heidelberg 2012

239

240 References

17. Ch. Antonopoulos, V. Basios, T. Bountis, Weak chaos and the “melting transition” in aconfined microplasma system. Phys. Rev. E. 81, 016211 (2010)

18. Ch. Antonopoulos, T. Bountis, V. Basios, Quasi-stationary chaotic states of multidimensionalHamiltonian systems. Phys. A 390, 3290–3307 (2011)

19. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989)20. V.I. Arnold, A. Avez, Problemes Ergodiques de la Mecanique Classique (Gauthier-Villars,

Paris, 1967 / Benjamin, New York, 1968)21. S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization. Phys. D

103, 201–250 (1997)22. F. Baldovin, E. Brigatti, C. Tsallis, Quasi-stationary states in low-dimensional Hamiltonian

systems. Phys. Lett. A 320, 254–260 (2004)23. F. Baldovin, L.G. Moyano, A.P. Majtey, A. Robledo, C. Tsallis, Ubiquity of metastable-to-

stable crossover in weakly chaotic dynamical systems. Phys. A 340, 205–218 (2004)24. D. Bambusi, A. Ponno, On metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006)25. D. Bambusi, A. Ponno, Resonance, Metastability and Blow Up in FPU. Lecture Notes in

Physics, vol. 728 (Springer, New York/Berlin, 2008), pp. 191–20526. R. Barrio, Sensitivity tools vs. Poincare sections. Chaos Soliton Fract. 25, 711–726 (2005)27. R. Barrio, Painting chaos: a gallery of sensitivity plots of classical problems.

Int. J. Bifurc. Chaos 16, 2777–2798 (2006)28. R. Barrio, W. Borczyk, S. Breiter, Spurious structures in chaos indicators maps. Chaos Soliton

Fract. 40, 1697–1714 (2009)29. C. Beck, Brownian motion from deterministic dynamics. Phys. A 169, 324–336 (1990)30. G. Benettin, A. Ponno, Time-scales to equipartition in the Fermi-Pasta-Ulam problem: finite

size effects and thermodynamic limit. J. Stat. Phys. 144, 793–812 (2011)31. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for

smooth dynamical systems and for Hamiltonian systems; a method for computing all of them.Part 1: Theory. Meccanica 15, 9–20 (1980)

32. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents forsmooth dynamical systems and for Hamiltonian systems; a method for computing all of them.Part 2: Numerical application. Meccanica 15, 21–30 (1980)

33. G. Benettin, L. Galgani, A. Giorgilli, A proof of Nekhoroshev’s theorem for the stability timesin nearly integrable Hamiltonian systems. Celest. Mech. 37, 1–25 (1985)

34. G. Benettin, A. Carati, L. Galgani, A. Giorgilli, The Fermi-Pasta-Ulam problem and themetastability perspective. Lecture Notes in Physics, vol. 728 (Springer, New York/Berlin,2008), pp. 151–189

35. G. Benettin, R. Livi, A. Ponno, The Fermi-Pasta-Ulam problem: scaling laws vs. initialconditions. J. Stat. Phys. 135, 873–893 (2009)

36. D. Benisti, D.F. Escande, Nonstandard diffusion properties of the standard map. Phys. Rev.Lett. 80, 4871–4874 (1998)

37. L. Berchialla, A. Giorgilli, S. Paleari, Exponentially long times to equipartition in thethermodynamic limit. Phys. Lett. A, 321, 167–172 (2004)

38. L. Berchialla, L. Galgani, A. Giorgilli, Localization of energy in FPU chains. Discret.Contin. Dyn. Syst. 11, 855–866 (2004)

39. J.M. Bergamin, Numerical approximation of breathers in lattices with nearest-neighborinteractions, Phys. Rev. E 67, 026703 (2003)

40. J.M. Bergamin, Localization in nonlinear lattices and homoclinic dynamics. Ph.D. Thesis,University of Patras, 2003

41. J.M. Bergamin, T. Bountis, C. Jung, A method for locating symmetric homoclinic orbits usingsymbolic dynamics. J. Phys. A-Math. Gen. 33, 8059–8070 (2000)

42. J.M. Bergamin, T. Bountis, M.N. Vrahatis, Homoclinic orbits of invertible maps. Nonlinearity15, 1603–1619 (2002)

43. G.P. Berman, F.M. Izrailev, The Fermi-Pasta-Ulam problem: fifty years of progress. Chaos15, 015104 (2005)

44. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)

References 241

45. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, A. Aspect, Direct observation of Anderson localization of matter-wavesin a controlled disorder. Nature 453, 891–894 (2008)

46. G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Wiley, New York, 1978)47. J.D. Bodyfelt, T.V. Laptyeva, Ch. Skokos, D.O. Krimer, S. Flach, Nonlinear waves in

disordered chains: probing the limits of chaos and spreading. Phys. Rev. E 84, 016205 (2011)48. J.D. Bodyfelt, T.V. Laptyeva, G. Gligoric, D.O. Krimer, Ch. Skokos, S. Flach, Wave

interactions in localizing media – a coin with many faces. Int. J. Bifurc. Chaos 21, 2107–2124 (2011)

49. J. Boreux, T. Carletti, Ch. Skokos, M. Vittot, Hamiltonian control used to improve the beamstability in particle accelerator models. Commun. Nonlinear Sci. Numer. Simul. (2011) 17,1725–1738 (2012)

50. J. Boreux, T. Carletti, Ch. Skokos, Y. Papaphilippou, M. Vittot, Efficient control of acceleratormaps. Int. J. Bifurc. Chaos (2012, In Press) E-print arXiv:1103.5631

51. T. Bountis, Investigating non-integrability and Chaos in complex time. Phys. D 86, 256–267(1995)

52. T. Bountis, Stability of motion: From Lyapunov to the dynamics N-degree of freedomHamiltonian systems. Nonlinear Phenomena and Complex Systems 9, 209–239 (2006)

53. T. Bountis, J.M. Bergamin, Discrete Breathers in Nonlinear Lattices: A Review and RecentResults. Lecture Notes in Physics, vol. 626 (Springer, New York/Berlin, 2003)

54. T. Bountis, M. Kollmann, Diffusion rates in a 4-dimensional mapping model of acceleratordynamics. Phys. D 71, 122–131 (1994)

55. T. Bountis, K.E. Papadakis, The stability of vertical motion in the N-body circular Sitnikovproblem. Celest. Mech. Dyn. Astron. 104, 205–225 (2009)

56. T. Bountis, H. Segur, in Logarithmic Singularities and Chaotic Behavior in HamiltonianSystems, ed. by M. Tabor, Y. Treves. A.I.P. Conference Proceedings, vol. 88, 279–292 (A.I.P.,New York, 1982)

57. T. Bountis, Ch. Skokos, Application of the SALI chaos detection method to acceleratormappings. Nucl. Instrum. Methods A 561, 173–179 (2006)

58. T. Bountis, Ch. Skokos, Space charges can significantly affect the dynamics of acceleratormaps. Phys. Lett. A 358, 126–133 (2006)

59. T. Bountis, S. Tompaidis, Strong and weak instabilities in a 4-D mapping model of acceleratordynamics, in Nonlinear Problems in Future Particle Accelerators, ed. by W. Scandale,G. Turchetti (World Scientific, Singapore, 1991), pp. 112–127

60. T. Bountis, H. Segur, F. Vivaldi, Integrable Hamiltonian systems and the Painleve property.Phys. Rev. A 25, 1257–1264 (1982)

61. T. Bountis, H.W. Capel, M. Kollmann, J.C. Ross, J.M. Bergamin, J.P. van der Weele,Multibreathers and homoclinic orbits in one-dimensional nonlinear lattices. Phys. Lett. A268, 50–60 (2000)

62. T. Bountis, J.M. Bergamin, V. Basios, Stabilization of discrete breathers using continuousfeedback control. Phys. Lett. A 295, 115–120 (2002)

63. T. Bountis, T. Manos, H. Christodoulidi, Application of the GALI method to localizationdynamics in nonlinear systems. J. Comput. Appl. Math. 227, 17–26 (2009)

64. T. Bountis, G. Chechin, V. Sakhnenko, Discrete symmetries and stability in Hamiltoniandynamics. Int. J. Bifurc. Chaos 21, 1539–1582 (2011)

65. V.A. Brazhnyi, V.V. Konotop, Theory of nonlinear matter waves in optical lattices. Mod. Phys.Lett. B 18, 627–651 (2004)

66. N. Budinsky, T. Bountis, Stability of nonlinear modes and chaotic properties of 1D Fermi-Pasta-Ulam lattices. Phys. D 8, 445–452 (1983)

67. A. Cafarella, M. Leo, R.A. Leo, Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system. Phys. Rev. E 69, 046604 (2004)

68. P. Calabrese, A. Gambassi, Slow dynamics in critical ferromagnetic vector models relaxingfrom a magnetized initial state. J. Stat. Mech.-Theory Exp. 2007, P01001 (2007)

242 References

69. F. Calogero, D. Gomez-Ullate, P.M. Santini, M. Sommacal, On the transition from regular toirregular motions, explained as travel on Riemann surfaces. J. Phys. A 38, 8873–8896 (2005)

70. F. Calogero, D. Gomez-Ullate, P.M. Santini, M. Sommacal, Towards a theory of chaosexplained as travel on Riemann surfaces. J. Phys. A 42, 015205 (2009)

71. D.K. Campbell, P. Rosenau, G.M. Zaslavsky (eds.), The Fermi-Pasta-Ulam problem: the first50 Years. Chaos, Focus Issue 15, 015101 (2005)

72. R. Capuzzo-Dolcetta, L. Leccese, D. Merritt, A. Vicari, Self-consistent models of cuspytriaxial galaxies with dark matter haloes. Astrophys. J. 666, 165–180 (2007)

73. J.R. Cary, D.F. Escande, A.D. Verga, Nonquasilinear diffusion far from the chaotic threshold.Phys. Rev. Lett. 65, 3132–3135 (1990)

74. G. Casati, B. Li, Heat conduction in one dimensional systems: Fourier law, chaos, andheat control, in Nonlinear Dynamics and Fundamental Interactions. NATO Science Series,Springer, New York/Berlin, vol. 213, Part 1, 1–16 (2006)

75. G. Casati, T. Prosen, Mixing property of triangular billiards. Phys. Rev. Lett. 83, 4729–4732(1999)

76. G. Casati, J. Ford, F. Vivaldi, W.M. Visscher, One-dimensional classical many-body systemhaving a normal thermal conductivity. Phys. Rev. Lett. 52, 1861–1864 (1984)

77. A. Celikoglu, U. Tirnakli, S.M. Duarte Queiros, Analysis of return distributions in thecoherent noise model. Phys. Rev. E 82, 021124 (2010)

78. J. Chabe, G. Lemarie, B. Gremaud, D. Delande, P. Szriftgiser, J.-C. Garreau, Experi-mental observation of the Anderson metal-insulator transition with atomic matter waves.Phys. Rev. Lett. 101, 255702 (2008)

79. C.W. Chang, D. Okawa, A. Majumdar, A. Zettl, Solid-state thermal rectifier. Science 314,1121 (2006)

80. G.M. Chechin, Computers and group-theoretical methods for studying structural phasetransition. Comput. Math. Appl. 17, 255–278 (1989)

81. G.M. Chechin, V.P. Sakhnenko, Interactions between normal modes in nonlinear dynamicalsystems with discrete symmetry. Exact results. Phys. D 117, 43–76 (1998)

82. G.M. Chechin, K.G. Zhukov, Stability analysis of dynamical regimes in nonlinear systemswith discrete symmetries. Phys. Rev. E 73, 036216 (2006)

83. G.M. Chechin, T.I. Ivanova, V.P. Sakhnenko, Complete order parameter condensate of low-symmetry phases upon structural phase transitions. Phys. Status Solidi B 152, 431–446 (1989)

84. G.M. Chechin, E.A. Ipatova, V.P. Sakhnenko, Peculiarities of the low-symmetry phasestructure near the phase-transition point. Acta Crystallogr. A 49, 824–831 (1993)

85. G.M. Chechin, N.V. Novikova, A.A. Abramenko, Bushes of vibrational modes for Fermi-Pasta-Ulam chains. Phys. D 166, 208–238 (2002)

86. G.M. Chechin, A.V. Gnezdilov, M.Yu. Zekhtser, Existence and stability of bushes of vibra-tional modes for octahedral mechanical systems with Lennard-Jones potential. Int. J. Nonlin-ear Mech. 38, 1451–1472 (2003)

87. G.M. Chechin, D.S. Ryabov, K.G. Zhukov, Stability of low-dimensional bushes of vibrationalmodes in the Fermi-Pasta-Ulam chains. Phys. D 203, 121–166 (2005)

88. B.V. Chirikov, A universal instability of many-dimensional oscillator systems. Phys. Rep. 52,263–379 (1979)

89. B.V. Chirikov, D.L. Shepelyansky, Correlation properties of dynamical chaos in Hamiltoniansystems. Phys. D 13, 395–400 (1984)

90. S.-N. Chow, M. Yamashita, Geometry of the Melnikov vector, in Nonlinear Equations inApplied Sciences, ed. by W.F. Ames, C. Rogers (Academic Press, San Diego, 1991), pp. 79–148

91. D.N. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behavious in linear andnonlinear waveguide lattices. Nature 424, 817 (2003)

92. H. Christodoulidi, Dynamics on low-dimensional tori and chaos in Hamiltonian systems.Ph.D. Thesis, University of Patras, 2010

93. H. Christodoulidi, T. Bountis, Low-dimensional quasiperiodic motion in Hamiltonian sys-tems. ROMAI J. 2, 37–44 (2006)

References 243

94. H. Christodoulidi, C. Efthymiopoulos, T. Bountis, Energy localization on q-tori, long-termstability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys. Rev. E 81, 016210(2010)

95. P.M. Cincotta, C. Simo, Simple tools to study global dynamics in non-axisymmetric galacticpotentials-I. Astron. Astrophys. Suppl. 147, 205–228 (2000)

96. P.M. Cincotta, C.M. Giordano, C. Simo, Phase space structure of multi-dimensional systemsby means of the mean exponential growth factor of nearby orbits. Phys. D 182, 151–178(2003)

97. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill,New York, 1955)

98. R.M. Conte, M. Musette, The Painleve Handbook (Springer, Heidelberg, 2008)99. G. Contopoulos, Order and Chaos in Dynamical Astronomy (Springer, Heidelberg, 2002)

100. G. Contopoulos, B. Barbanis, Lyapunov characteristic numbers and the structure of phase-space. Astron. Astrophys. 222, 329–343 (1989)

101. G. Contopoulos, P. Magnenat, Simple three-dimensional periodic orbits in a galactic-typepotential. Celest. Mech. 37, 387–414 (1985)

102. G. Contopoulos, N. Voglis, Spectra of stretching numbers and helicity angles in dynamicalsystems. Celest. Mech. Dyn. Astr. 64, 1–20 (1996)

103. G. Contopoulos, N. Voglis, A fast method for distinguishing between ordered and chaoticorbits. Astron. Astrophys. 317, 73–81 (1997)

104. G. Contopoulos, L. Galgani, A. Giorgilli, On the number of isolating integrals in Hamiltoniansystems. Phys. Rev. A 18, 1183–1189 (1978)

105. T. Cretegny, T. Dauxois, S. Ruffo, A. Torcini, Localization and equipartition of energy in thebeta-FPU chain: chaotic breathers. Phys. D 121, 109–126 (1998)

106. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation intrapped gases. Rev. Mod. Phys. 71, 463–512 (1999)

107. H.T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York,1962)

108. T. Dauxois, Non-Gaussian distributions under scrutiny. J. Stat. Mech.-Theory Exp. 2007,N08001 (2007)

109. J. De Luca, A.J. Lichtenberg, Transitions and time scales to equipartition in oscillator chains:low-frequency initial conditions. Phys. Rev. E 66, 026206 (2002)

110. J. De Luca, A.J. Lichtenberg, M.A. Lieberman, Time scale to ergodicity in the Fermi-Pasta-Ulam system. Chaos 5, 283–297 (1995)

111. J. De Luca, A.J. Lichtenberg, S. Ruffo, Energy transitions and time scales to equipartition inthe Fermi-Pasta-Ulam oscillator chain. Phys. Rev. E 51, 2877–2885 (1995)

112. J. De Luca, A.J. Lichtenberg, S. Ruffo, Finite times to equipartition in the thermodynamiclimit. Phys. Rev. E 60, 3781–3786 (1999)

113. L. Drossos, T. Bountis, Evidence of natural boundary and nonintegrability of the mixmasteruniverse model. J. Nonlinear Sci. 7, 1–11 (1997)

114. W.E. Drummond, D. Pines, Nonlinear stability of plasma oscillations. Nucl. Fusion Suppl. 3,1049–1057 (1962)

115. S.M. Duarte Queiros, The role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables. Phys. Lett. A 373, 1514–1518 (2009)

116. G. Duffing, Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technischeBedeutung (Vieweg & Sohn, Braunschweig, 1918)

117. J.-P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57,617–656 (1985)

118. J.T. Edwards, D.J. Thouless, Numerical studies of localization in disordered systems.J. Phys. C Solid 5, 807–820 (1972)

119. N.K. Efremidis, D.N. Christodoulides, Lattice solitons in Bose-Einstein condensates. Phys.Rev. A 67, 063608 (2003)

120. L.H. Eliasson, Absolutely convergent series expansions for quasi periodic motions.Math. Phys. Electron. J. 2, 4 (1996)

244 References

121. E. Fermi, J. Pasta, S. Ulam, Studies of nonlinear problems. Los AlamosSci. Lab. Rep. No. LA-1940 (1955), in Nonlinear Wave Motion, ed. by A.C. Newell.Lectures in Applied Mathematics, vol. 15 (Amer. Math. Soc., Providence, 1974),pp. 143–155

122. S. Flach, Conditions on the existence of localized excitations in nonlinear discrete systems.Phys. Rev. E 50, 3134–3142 (1994)

123. S. Flach, Obtaining breathers in nonlinear Hamiltonian lattices. Phys. Rev. E 51, 3579–3587(1995)

124. S. Flach, Spreading of waves in nonlinear disordered media. Chem. Phys. 375, 548–556(2010)

125. S. Flach, A.V. Gorbach, Discrete breathers – Advances in theory and applications. Phys. Rep.467, 1–116 (2008)

126. S. Flach, A. Ponno, The Fermi-Pasta-Ulam problem: periodic orbits, normal forms andresonance overlap criteria. Phys. D 237, 908–917 (2008)

127. S. Flach, C. Willis, Discrete breathers. Phys. Rep. 295, 181–264 (1998)128. S. Flach, M.V. Ivanchenko, O.I. Kanakov, q-Breathers and the Fermi-Pasta-Ulam problem.

Phys. Rev. Lett. 95, 064102 (2005)129. S. Flach, M.V. Ivanchenko, O.I. Kanakov, q-breathers in Fermi-Pasta-Ulam chains: existence,

localization, and stability. Phys. Rev. E 73, 036618 (2006)130. S. Flach, O.I. Kanakov, M.V. Ivanchenko, K. Mishagin, q-breathers in FPU-lattices – scaling

and properties for large systems. Int. J. Mod. Phys. B 21, 3925–3932 (2007)131. S. Flach, D.O. Krimer, Ch. Skokos, Universal spreading of wavepackets in disordered

nonlinear systems. Phys. Rev. Lett. 102, 024101 (2009)132. A.S. Fokas, T. Bountis, Order and the ubiquitous occurrence of Chaos. Phys. A 228, 236–244

(1996)133. J. Ford, The Fermi-Pasta-Ulam problem: paradox turns discovery. Phys. Rep. 213, 271–310

(1992)134. F. Freistetter, Fractal dimensions as chaos indicators. Celest. Mech. Dyn. Astron. 78, 211–225

(2000)135. C. Froeschle, E. Lega, On the structure of symplectic mappings. The fast Lyapunov indicator:

A very sensitive tool. Celest. Mech. Dyn. Astron. 78, 167–195 (2000)136. C. Froeschle, Ch. Froeschle, E. Lohinger, Generalized Lyapunov characteristic indicators and

corresponding Kolmogorov like entropy of the standard mapping. Celest. Mech. Dyn. Astron.56, 307–314 (1993)

137. C. Froeschle, E. Lega, R. Gonczi, Fast Lyapunov indicators. Application to asteroidal motion.Celest. Mech. Dyn. Astron. 67, 41–62 (1997)

138. C. Froeschle, R. Gonczi, E. Lega, The fast Lyapunov indicator: a simple tool to detect weakchaos. Application to the structure of the main asteroidal belt. Planet. Space Sci. 45, 881–886(1997)

139. F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo, A. Vulpiani, Approach toequilibrium in a chain of nonlinear oscillators. J. Phys.-Paris 43, 707–713 (1982)

140. L. Galgani, A. Scotti, Planck-like distributions in classical nonlinear mechanics.Phys. Rev. Lett. 28, 1173–1176 (1972)

141. Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic. Nonlin-earity 15, 1759–1779 (2002)

142. G. Gallavotti, Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994)143. G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations

in the perturbation series of certain completely integrable Hamiltonian systems: a review.Rev. Math. Phys. 6, 343–411 (1994)

144. I. Garcıa-Mata, D.L. Shepelyansky, Delocalization induced by nonlinearity in systems withdisorder. Phys. Rev. E 79, 026205 (2009)

145. P. Gaspard, Lyapunov exponent of ion motion in microplasmas. Phys. Rev. E 68, 056209(2003)

References 245

146. E. Gerlach, Ch. Skokos, Comparing the efficiency of numerical techniques for the integrationof variational equations. Discr. Cont. Dyn. Sys.-Supp. September, 475–484 (2011)

147. E. Gerlach, S. Eggl, Ch. Skokos, Efficient integration of the variational equationsof multi-dimensional Hamiltonian systems: application to the Fermi-Pasta-Ulam lattice.Int. J. Bifurc. Chaos (2012, In Press) E-print arXiv:1104.3127

148. A. Giorgilli, U. Locatelli, Kolmogorov theorem and classical perturbation theory.Z. Angew. Math. Phys. 48, 220–261 (1997)

149. A. Giorgilli, U. Locatelli, A classical self-contained proof of Kolmogorov’s theorem oninvariant tori, in Hamiltonian Systems of Three or More Degrees of Freedom, ed. by C. Simo.NATO Advanced Study Institute, vol. 533 (Kluwer, Dordrecht, 1999), pp. 72–89

150. A. Giorgilli, D. Muraro, Exponentially stable manifolds in the neighbourhood of ellipticequilibria. Boll. Unione Mate. Ital. B 9, 1–20 (2006)

151. M.L. Glasser, V.G. Papageorgiou, T.C. Bountis, Mel’nikov’s function for two-dimensionalmappings. SIAM J. Appl. Math. 49, 692–703 (1989)

152. A. Goriely, Integrability and Nonintegrability of Dynamical Systems (World Scientific,Singapore, 2001)

153. G.A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems.Proc. R. Soc. Lond. A Math. 460, 603–611 (2004)

154. G.A. Gottwald, I. Melbourne, Testing for chaos in deterministic systems with noise. Phys. D212, 100–110 (2005)

155. E. Goursat, Cours d’ Analyse Mathematique vol. 2 (Gauthier-Villars, Paris, 1905)156. B. Grammaticos, B. Dorizzi, R. Padjen, Painleve property and integrals of motion for the

Henon-Heiles system. Phys. Lett. A 89, 111–113 (1982)157. P.E. Greenwood, M.S. Nikulin, A Guide to Chi-Squared Testing, (Wiley, New York, 1996)158. P. Grassberger, Proposed central limit behavior in deterministic dynamical systems.

Phys. Rev. E 79, 057201 (2009)159. W. Greub, Multilinear Algebra, 2nd edn. (Springer, Heidelberg, 1978)160. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of

Vector Fields (Springer, New York, 1983)161. M.G. Hahn, X. Jiang, S. Umarov, On q-Gaussians and exchangeability. J. Phys.

A-Math. Theor. 43, 165208 (2010)162. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving

Algorithms for Ordinary Differential Equations. Springer Series in Comput. Math., vol. 31(Springer, Berlin, 2002)

163. P. Hemmer, Dynamic and stochastic type of motion by the linear chain. Det Physiske Seminari Trondheim 2, 66 (1959)

164. M. Henon, C. Heiles, The applicability of the third integral of motion: some numericalexperiments. Astron. J. 69, 73–79 (1964)

165. R.C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 1994)166. H.J. Hilhorst, Note on a q-modified central limit theorem. J. Stat. Mech.-Theory Exp. 2010,

P10023 (2010)167. H.J. Hilhorst, G. Schehr, A note on q-Gaussians and non-Gaussians in statistical mechanics.

J. Stat. Mech.-Theory Exp. 2007, P06003 (2007)168. T.L. Hill Thermodynamics of Small Systems (Dover, New York, 1994)169. E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, 1969)170. M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems and an

Introduction to Chaos (Elsevier, New York, 2004)171. J.E. Howard, Discrete virial theorem. Celest. Mech. Dyn. Astron. 92, 219–241 (2005)172. B. Hu, B. Li, H. Zhao, Heat conduction in one-dimensional chains. Phys. Rev. E 57, 2992

(1998)173. H. Hu, A. Strybulevych, J. Page, S. Skipetrov, B. van Tiggelen, Localization of ultrasound in

a three-dimensional elastic network. Nat. Phys. 4, 945–948 (2008)174. J.H. Hubbard, B.B. Hubbard, Vector Calculus, Linear Algebra and Differential Forms:

A Unified Approach (Prentice Hall, Upper Saddle River, 1999)

246 References

175. M.C. Irwin, Smooth Dynamical Systems (Academic, New York, 1980)176. N. Jacobson, Lectures in Abstract Algebra, vol. II (van Nostrand, Princeton, 1951)177. M. Johansson, G. Kopidakis, S. Lepri, S. Aubry, Transmission thresholds in time-periodically

driven nonlinear disordered systems. Europhys. Lett. 86, 10009 (2009)178. M. Johansson, G. Kopidakis, S. Aubry, KAM tori in 1D random discrete nonlinear

Schrodinger model? Europhys. Lett. 91, 50001 (2010)179. O.I. Kanakov, S. Flach, M.V. Ivanchenko, K.G. Mishagin, Scaling properties of q-breathers

in nonlinear acoustic lattices. Phys. Lett. A 365, 416–420 (2007)180. H. Kantz, P. Grassberger, Internal Arnold diffusion and chaos thresholds in coupled symplec-

tic maps. J. Phys. A-Math. Gen. 21, L127–L133 (1988)181. G.I. Karanis, Ch.L. Vozikis, Fast detection of chaotic behavior in galactic potentials.

Astron. Nachr. 329, 403–412 (2008)182. Y.V. Kartashov, V.A. Vysloukh, L. Torner, Soliton shape and mobility control in optical

lattices. Prog. Opt. 52, 63–148 (2009)183. Y.V. Kartashov, B.A. Malomed, L. Torner, Solitons in nonlinear lattices. Rev. Mod. Phys. 83,

247 (2011)184. A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.

Publ. Math. IHES 51, 137–173 (1980)185. A. Katok, J.-M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with

Singularities. Lecture Notes in Mathematics, vol. 1222 (Springer, Berlin, 1986)186. A.N. Kaufman, Quasilinear diffusion of an axisymmetric toroidal plasma. Phys. Fluids 15,

1063 (1972)187. W. Ketterle, D.S. Durfee, D.M. Stamper-Kurn, Making, probing and understanding Bose-

Einstein condensates, in Bose-Einstein Condensation in Atomic Gases. Proceedings ofthe International School of Physics “Enrico Fermi”, ed. by M. Inguscio, S. Stringari,C.E. Wieman (IOS Press, Amsterdam, 1999), pp. 67–176

188. P.G. Kevrekidis, The Discrete Nonlinear Schrodinger Equation. Tracts in Modern Physics,vol. 232 (Springer, Heidelberg, 2009)

189. Y.S. Kivshar, Intrinsic localized modes as solitons with a compact support. Phys. Rev. E 48,R43–R45 (1993)

190. Y.S. Kivshar, G.P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Academic,Amsterdam, 2003)

191. Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in near-integrable systems. Rev. Mod.Phys. 61, 763–915 (1989)

192. W. Kobayashi, Y. Teraoka, I. Terasaki, An oxide thermal rectifier. Appl. Phys. Lett. 95, 171905(2009)

193. Y. Kominis, Analytical solitary wave solutions of the nonlinear Kronig-Penney model inphotonic structures. Phys. Rev. E 73, 066619 (2006)

194. Y. Kominis, T. Bountis, Analytical solutions of systems with piecewise linear dynamics. Int.J. Bifurc. Chaos 20, 509–518 (2010)

195. Y. Kominis, K. Hizanidis, Lattice solitons in self-defocusing optical media: analyticalsolutions of the nonlinear Kronig-Penney model. Opt. Lett. 31, 2888–2890 (2006)

196. Y. Kominis, K. Hizanidis, Power dependent soliton location and stability in complex photonicstructures. Opt. Expr. 16, 12124–12138 (2008)

197. Y. Kominis, K. Hizanidis, Power-dependent reflection, transmission and trapping dynamicsof lattice solitons at interfaces. Phys. Rev. Lett. 102, 133903 (2009)

198. Y. Kominis, A. Papadopoulos, K. Hizanidis, Surface solitons in waveguide arrays: analyticalsolutions. Opt. Expr. 15, 10041–10051 (2007)

199. Y. Kominis, A.K. Ram, K. Hizanidis, Quasilinear theory of electron transport by radiofrequency waves and non-axisymmetric perturbations in toroidal plasmas. Phys. Plasmas 15,122501 (2008)

200. Y. Kominis, T. Bountis, K. Hizanidis, Breathers in a nonautonomous Toda lattice withpulsating coupling. Phys. Rev. E 81, 066601 (2010)

References 247

201. Y. Kominis, A.K. Ram, K. Hizanidis, Kinetic theory for distribution functions of wave-particle interactions in plasmas. Phys. Rev. Lett. 104, 235001 (2010)

202. G. Kopidakis, S. Komineas, S. Flach, S. Aubry, Absence of wave packet diffusion indisordered nonlinear systems. Phys. Rev. Lett. 100, 084103 (2008)

203. Y.A. Kosevich, Nonlinear sinusoidal waves and their superposition in anharmonic lattices.Phys. Rev. Lett. 71, 2058–2061 (1993)

204. T. Kotoulas, G. Voyatzis, Comparative study of the 2:3 and 3:4 resonant motionwith Neptune: an application of symplectic mappings and low frequency analysis.Celest. Mech. Dyn. Astron. 88, 343–363 (2004)

205. I. Kovacic, M.J. Brennan (eds.), The Duffing Equation: Nonlinear Oscillators and TheirBehaviour (Wiley, Hoboken, 2011)

206. B. Kramer, A. MacKinnon, Localization: theory and experiment. Rep. Prog. Phys. 56, 1469–1564 (1993)

207. D.O. Krimer, S. Flach, Statistics of wave interactions in nonlinear disordered systems.Phys. Rev. E 82, 046221 (2010)

208. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, Y. Silberberg,Anderson localization and nonlinearity in one-dimensional disordered photonic lattices.Phys. Rev. Lett. 100, 013906 (2008)

209. L.D. Landau, E.M. Lifshitz, Mechanics, Third edn, Volume 1 of Course of Theoretical Physics(Butterworth-Heinemann, Amsterdam, 1976)

210. T.V. Laptyeva, J.D. Bodyfelt, D.O. Krimer, Ch. Skokos, S. Flach, The crossover from strongto weak chaos for nonlinear waves in disordered systems. Europhys. Lett. 91, 30001 (2010)

211. J. Laskar, The chaotic motion of the Solar System: a numerical estimate of the size of thechaotic zones. Icarus 88, 266–291 (1990)

212. J. Laskar, Frequency analysis of multi-dimensional systems. Global dynamics and diffusion.Phys. D 67, 257–281 (1993)

213. J. Laskar, Introduction to frequency map analysis, in Hamiltonian Systems of Three or MoreDegrees of Freedom, ed. by C. Simo. NATO Advanced Study Institute, vol. 533 (Kluwer,Dordrecht, 1999), pp. 134–150

214. J. Laskar, C. Froeschle, A. Celletti, The measure of chaos by the numerical analysis of thefundamental frequencies. Application to the standard map. Phys. D 56, 253–269 (1992)

215. M. Lax, W.H. Louisell, W.B. McKnight, From Maxwell to paraxial wave optics. Phys. Rev.A 11, 1365–1370 (1975)

216. F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg,Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008)

217. E. Lega, C. Froeschle, Comparison of convergence towards invariant distributions for rotationangles, twist angles and local Lyapunov characteristic numbers. Planet. Space Sci. 46, 1525–1534 (1998)

218. M. Leo, R.A. Leo, Stability properties of the N=4 (�=2-mode) one-mode nonlinear solutionof the Fermi-Pasta-Ulam-ˇ system. Phys. Rev. E 76, 016216 (2007)

219. S. Lepri, R. Livi, A. Politi, Thermal conduction in classical low-dimensional lattices. Phys.Rep. 377(1), 1–80 (2003)

220. S. Lepri, R. Livi, A. Politi, Studies of thermal conductivity in Fermi Pasta Ulam-like lattices.Chaos 15, 015118 (2005)

221. B. Li, J. Wang, Anomalous heat conduction and anomalous diffusion in one-dimensionalsystems. Phys. Rev. Lett. 91, 044301 (2003)

222. B. Li, L. Wang, B. Hu, Finite thermal conductivity in 1D models having zero Lyapunovexponents. Phys. Rev. Lett. 88, 223901 (2002)

223. B. Li, G. Casati, J. Wang, Heat conductivity in linear mixing systems. Phys. Rev. E 67, 021204(2003)

224. B. Li, G. Casati, J. Wang, T. Prosen, Fourier Law in the alternate-mass hard-core potentialchain. Phys. Rev. Lett. 92, 254301 (2004)

225. B. Li, J. Wang, G. Casati, Thermal diode: rectification of heat flux. Phys. Rev. Lett. 93, 184301(2004)

248 References

226. A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, Second edn. (Springer,New York, 1992)

227. A. Lichtenberg, R. Livi, M. Pettini, S. Ruffo, Dynamics of oscillator chains. Lect. Notes Phys.728, 21–121 (2008)

228. R. Livi, M. Pettini, S. Ruffo, A. Vulpiani, Further results on the equipartition threshold inlarge nonlinear Hamiltonian systems. Phys. Rev. A 31, 2740–2742 (1985)

229. R. Livi, A. Politi, S. Ruffo, Distribution of characteristic exponents in the thermodynamiclimit. J. Phys. A-Math. Gen. 19, 2033–2040 (1986)

230. W.C. Lo, L. Wang, B. Li, Thermal transistor: heat flux switching and modulating. J. Phys.Soc. Jpn, 77(5), 054402 (2008)

231. E. Lohinger, C. Froeschle, R. Dvorak, Generalized Lyapunov exponents indicators inHamiltonian dynamics: an application to a double star system. Celest. Mech. Dyn. Astron.56, 315–322 (1993)

232. A.M. Lyapunov, The General Problem of the Stability of Motion (Taylor and Francis, London,1992) (English translation from the French: A. Liapounoff, Probleme general de la stabilitedu mouvement. Annal. Fac. Sci. Toulouse 9, 203–474 (1907). The French text was reprintedin Annals Math. Studies Vol.17 Princeton Univ. Press (1947). The original was published inRussian by the Mathematical Society of Kharkov in 1892)

233. M. Macek, P. Stransky, P. Cejnar, S. Heinze, J. Jolie, J. Dobes, Classical and quantumproperties of the semiregular arc inside the Casten triangle. Phys. Rev. C 75, 064318 (2007)

234. M. Macek, J. Dobes, P. Stransky, P. Cejnar, Regularity-induced separation of intrinsic andcollective dynamics. Phys. Rev. Lett. 105, 072503 (2010)

235. R.S. MacKay, S. Aubry, Proof of existence of breathers for time-reversible or Hamiltoniannetworks of weakly coupled oscillators. Nonlinearity 7, 1623–1843 (1994)

236. R.S. Mackay, J.D. Meiss, Hamiltonian Dynamical Systems (Adam Hilger, Bristol, 1986)237. M.C. Mackey, M. Tyran-Kaminska, Deterministic Brownian motion: the effects of perturbing

a dynamical system by a chaotic semi-dynamical system. Phys. Rep. 422, 167–222 (2006)238. R.S. Mackay, J.D. Meiss, I.C. Percival, Transport in Hamiltonian systems. Phys. D 13, 55–81

(1984)239. N.P. Maffione, L.A. Darriba, P.M. Cincotta, C.M. Giordano, A comparison of different

indicators of chaos based on the deviation vectors: application to symplectic mappings.Celest. Mech. Dyn. Astron. 111, 285–307 (2011)

240. W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1969) and 2nd edn. (Dover,New York, 2004)

241. P. Maniadis, T. Bountis, Quasiperiodic and chaotic breathers in a parametrically driven systemwithout linear dispersion. Phys. Rev. E 73, 046211 (2006)

242. T. Manos, E. Athanassoula, Regular and chaotic orbits in barred galaxies – I. Applying theSALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc.415, 629–642 (2011)

243. T. Manos, S. Ruffo, Scaling with system size of the Lyapunov exponents for the HamiltonianMean Field model. Transp. Theor. Stat. 40, 360–381 (2011)

244. T. Manos, Ch. Skokos, T. Bountis, Application of the Generalized Alignment Index (GALI)method to the dynamics of multi-dimensional symplectic maps, in Chaos, Complexityand Transport: Theory and Applications. Proceedings of the CCT07, ed. by C. Chandre,X. Leoncini, G. Zaslavsky (World Scientific, Singapore, 2008), pp. 356–364

245. T. Manos, Ch. Skokos, E. Athanassoula, T. Bountis, Studying the global dynamics ofconservative dynamical systems using the SALI chaos detection method. Nonlinear Phe-nom. Complex Syst. 11, 171–176 (2008)

246. T. Manos, Ch. Skokos, T. Bountis, Global dynamics of coupled standard maps, in Chaos inAstronomy. Astrophysics and Space Science Proceedings, ed. by G. Contopoulos, P.A. Patsis(Springer, Berlin/Heidelberg, 2009), pp. 367–371

247. T. Manos, Ch. Skokos, Ch. Antonopoulos, Probing the local dynamics of periodic orbits bythe generalized alignment index (GALI) method. Int. J. Bifurc. Chaos (2012, In Press) E-printarXiv:1103.0700

References 249

248. J.L. Marın, S. Aubry, Breathers in nonlinear lattices: numerical calculation from the anticon-tinuous limit. Nonlinearity 9, 1501–1528 (1996)

249. J.D. Meiss, E. Ott, Markov tree model of transport in area-preserving maps. Phys. D 20, 387–402 (1986)

250. D.R. Merkin, Introduction to the Theory of Stability. Series: Texts in Applied Mathematics,vol. 24 (Springer, New York, 1997)

251. G. Miritello, A. Pluchino, A. Rapisarda, Central limit behavior in the Kuramoto model at the“edge of chaos”. Phys. A 388, 4818–4826 (2009)

252. M. Molina, Transport of localized and extended excitations in a nonlinear Anderson model.Phys. Rev. B 58, 12547–12550 (1998)

253. M. Mulansky, A. Pikovsky, Spreading in disordered lattices with different nonlinearities.Europhys. Lett. 90, 10015 (2010)

254. M. Mulansky, K. Ahnert, A. Pikovsky, D.L. Shepelyansky, Dynamical thermalization ofdisordered nonlinear lattices. Phys. Rev. E 80, 056212 (2009)

255. M. Mulansky, K. Ahnert, A. Pikovsky, Scaling of energy spreading in strongly nonlineardisordered lattices. Phys. Rev. E 83, 026205 (2011)

256. N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrableHamiltoninan systems. Russ. Math. Surv. 32(6), 1–65 (1977)

257. Z. Nitecki, Differentiable Dynamics (M.I.T., Cambridge, MA, 1971)258. J.A. Nunez, P.M. Cincotta, F.C. Wachlin, Information entropy. An indicator of chaos.

Celest. Mech. Dyn. Astron. 64, 43–53 (1996)259. V.I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynam-

ical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)260. E.A. Ostrovskaya, Y.S. Kivshar, Matter-wave gap vortices in optical lattices. Phys. Rev. Lett.

93, 160405 (2004)261. A.A. Ovchinnikov, Localized long-lived vibrational states in molecular crystals.

Sov. Phys. JETP-USSR 30, 147 (1970)262. P. Panagopoulos, T.C. Bountis, Ch. Skokos, Existence and stability of localized oscillations

in one-dimensional lattices with soft spring and hard spring potentials. J. Vib. Acoust. 126,520–527 (2004)

263. P. Papagiannis, Y. Kominis, K. Hizanidis, Power- and momentum-dependent soliton dynamicsin lattices with longitudinal modulation. Phys. Rev. A 84, 013820 (2011)

264. R.E. Peierls, Quantum theory of solids, in Theoretical Physics in the Twentieth Century, ed.by M. Fierz, V.F. Weisskopf (Wiley, New York, 1961) 140–160

265. L. Perko, Differential Equations and Dynamical Systems (Springer, New York, 1995)266. J.B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents.

Math. USSR Izv. 10, 1261–1305 (1976)267. Ya.B. Pesin, Lyapunov characteristic indexes and ergodic properties of smooth dynamic

systems with invariant measure. Dokl. Acad. Nauk. SSSR 226, 774–777 (1976)268. Ya.B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv.

32(4), 55–114 (1977)269. Y.G. Petalas, C.G. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Evolutionary methods for

the approximation of the stability domain and frequency optimization of conservative maps.Int. J. Bifurc. Chaos 18, 2249–2264 (2008)

270. M. Peyrard, The design of a thermal rectifier. Europhys. Lett. 76, 49 (2006)271. A. Pikovsky, D. Shepelyansky, Destruction of Anderson localization by a weak nonlinearity.

Phys. Rev. Lett. 100, 094101 (2008)272. P. Poggi, S. Ruffo, Exact solutions in the FPU oscillator chain. Phys. D 103, 251–272 (1997)273. H. Poincare, Sur les Proprietes des Functions Definies par les Equations aux Differences

Partielles (Gauthier-Villars, Paris, 1879)274. H. Poincare Les Methodes Nouvelles de la Mecanique Celeste, vol. 1 (Gauthier Villars, Paris,

1892) (English translation by D.L. Goroff, New Methods in Celestial Mechanics (AmericanInstitute of Physics, 1993))

250 References

275. A. Ponno, D. Bambusi, Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam.Chaos 15, 015107 (2005)

276. A. Ponno, E. Christodoulidi, Ch. Skokos, S. Flach, The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior. Chaos, 21, 043127 (2011)

277. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flanney, Numerical Recipes in Fortran 77.The Art of Scientific Computing, Second edn. (Cambridge University Press, Cambridge/NewYork, 2001)

278. K. Pyragas, Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170,421–428 (1992)

279. A. Ramani, B. Grammaticos, T. Bountis, The Painleve property and singularity analysis ofintegrable and non-integrable systems. Phys. Rep. 180, 159–245 (1989)

280. A.B. Rechester, R.B. White, Calculation of turbulent diffusion for the Chirikov-Taylor model.Phys. Rev. Lett. 44, 1586–1589 (1980)

281. A. Renyi, On measures of information and entropy, in Proceedings of the 4th BerkeleySymposium on Mathematics, Statistics and Probability 1960, University of California Press,Berkeley/Los Angeles, 1961, pp. 547–561

282. J.A. Rice, Mathematical Statistics and Data Analysis, Second edn. (Duxbury Press, Belmont,1995)

283. B. Rink, Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice. Phys. D 175, 31–42(2003)

284. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M.Modugno, M. Inguscio, Anderson localization of a non-interacting Bose-Einstein condensate.Nature 453, 895–899 (2008)

285. A. Rodrıguez, V. Schwammle, C. Tsallis, Strictly and asymptotically scale invariant prob-abilistic models of N correlated binary random variables having q-Gaussians as N ! 1limiting distributions. J. Stat. Mech.-Theory Exp. 2008, P09006 (2008)

286. R.M. Rosenberg, The normal modes of nonlinear n-degree-of-freedom systems.J. Appl. Mech. 29, 7–14 (1962)

287. V.M. Rothos, T. Bountis, Mel’nikov analysis of phase space transport in a N-degree-of-freedom Hamiltonian system. Nonlinear Anal. Theor. 30, 1365–1374 (1997)

288. V.M. Rothos, T. Bountis, Mel’nikov’s vector and singularity analysis of periodically perturbed2 d.o.f. Hamiltonian systems, in Hamiltonian Systems of Three or More Degrees of Freedom,ed. by C. Simo. NATO Advanced Study Institute, vol. 533 (Kluwer, Dordrecht, 1999),pp. 544–548

289. D. Ruelle, Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50, 27–58(1979)

290. D. Ruelle, Measures describing a turbulent flow. Ann. NY Acad.Sci. 357, 1–9 (1980)291. D. Ruelle, Large volume limit of the distribution of characteristic exponents in turbulence.

Commun. Math. Phys. 87, 287–302 (1982)292. G. Ruiz, C. Tsallis, Nonextensivity at the edge of chaos of a new universality class of one-

dimensional unimodal dissipative maps. Eur. Phys. J. B 67, 577–584 (2009)293. G. Ruiz, T. Bountis, C. Tsallis, Time-evolving statistics of chaotic orbits of conservative

maps in the context of the central limit theorem. Int. J. Bifurc. Chaos. (2012, In Press)arXiv:1106.6226

294. V.P. Sakhnenko, G.M. Chechin, Symmetrical selection rules in nonlinear dynamics of atomicsystems. Sov. Phys. Dokl. 38, 219–221 (1993)

295. V.P. Sakhnenko, G.M. Chechin, Bushes of modes and normal modes for nonlinear dynamicalsystems with discrete symmetry. Sov. Phys. Dokl. 39, 625–628 (1994)

296. Zs. Sandor, B. Erdi, C. Efthymiopoulos, The phase space structure around L4 in the restrictedthree-body problem. Celest. Mech. Dyn. Astron. 78, 113–123 (2000)

297. Zs. Sandor, B. Erdi, A. Szell, B. Funk, The relative Lyapunov indicator: an efficient methodof chaos detection. Celest. Mech. Dyn. Astron. 90, 127–138 (2004)

298. K.W. Sandusky, J.B. Page, Interrelation between the stability of extended normal modesand the existence of intrinsic localized modes in nonlinear lattices with realistic potentials.Phys. Rev. B 50, 866–887 (1994)

References 251

299. T. Schwartz, G. Bartal, S. Fishman, M. Segev, Transport and Anderson localization indisordered two-dimensional photonic lattices. Nature 446, 52–55 (2007)

300. H. Segur, M.D. Kruskal, Nonexistence of small-amplitude breather solutions in �4 theory.Phys. Rev. Lett. 58, 747–750 (1987)

301. V.D. Shapiro, R.Z. Sagdeev, Nonlinear wave-particle interaction and conditions for theapplicability of quasilinear theory. Phys. Rep. 283, 49–71 (1997)

302. H. Shiba, N. Ito, Anomalous heat conduction in three-dimensional nonlinear lattices. J. Phys.Soc. Jpn. 77, 05400 (2008)

303. S. Shinohara, Low-dimensional solutions in the quartic Fermi-Pasta-Ulam system.J. Phys. Soc. Jpn. 71, 1802–1804 (2002)

304. S. Shinohara, Low-dimensional subsystems in anharmonic lattices. Prog. Theor. Phys. Suppl.150, 423–434 (2003)

305. I.V. Sideris, Measure of orbital stickiness and chaos strength. Phys. Rev. E 73, 066217 (2006)306. A.J. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61,

970–973 (1988)307. Y.G. Sinai, Dynamical systems with elastic reflections. Russ. Math. Surv. 25(2), 137–189

(1970)308. Ya.G. Sinai, Gibbs measures in ergodic theory. Russ. Math. Surv. 27(4), 21–69 (1972)309. Ch. Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic

nature of orbits. J. Phys. A-Math. Gen. 34, 10029–10043 (2001)310. Ch. Skokos, The Lyapunov characteristic exponents and their computation. Lect. Notes Phys.

790, 63–135 (2010)311. Ch. Skokos, S. Flach, Spreading of wave packets in disordered systems with tunable

nonlinearity. Phys. Rev. E 82, 016208 (2010)312. Ch. Skokos, E. Gerlach, Numerical integration of variational equations. Phys. Rev. E 82,

036704 (2010)313. Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, How does the smaller alignment

index (SALI) distinguish order from chaos? Prog. Theor. Phys. Suppl. 150, 439–443 (2003)314. Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Detecting order and chaos in

Hamiltonian systems by the SALI method. J. Phys. A-Math. Gen. 37, 6269–6284 (2004)315. Ch. Skokos, T.C. Bountis, Ch. Antonopoulos, Geometrical properties of local dynamics in

Hamiltonian systems: the generalized alignment index (GALI) method. Phys. D 231, 30–54(2007)

316. Ch. Skokos, T. Bountis, Ch. Antonopoulos, Detecting chaos, determining the dimensions oftori and predicting slow diffusion in Fermi-Pasta-Ulam lattices by the generalized alignmentindex method. Eur. Phys. J.-Spec. Top. 165, 5–14 (2008)

317. Ch. Skokos, D.O. Krimer, S. Komineas, S. Flach, Delocalization of wave packets indisordered nonlinear chains. Phys. Rev. E 79, 056211 (2009)

318. A. Smerzi, A. Trombettoni, Nonlinear tight-binding approximation for Bose-Einstein con-densates in a lattice. Phys. Rev. A 68, 023613 (2003)

319. P. Soulis, T. Bountis, R. Dvorak, Stability of motion in the Sitnikov 3-body problem.Celest. Mech. Dyn. Astron. 99, 129–148 (2007)

320. P.S. Soulis, K.E. Papadakis, T. Bountis, Periodic orbits and bifurcations in the Sitnikov four-body problem. Celest. Mech. Dyn. Astron. 100, 251–266 (2008)

321. M. Spivak, Comprehensive Introduction to Differential Geometry, vol. 1 (Perish Inc.,Houston, 1999)

322. P. Stransky, P. Hruska, P. Cejnar, Quantum chaos in the nuclear collective model: classical-quantum correspondence. Phys. Rev. E 79, 046202 (2009)

323. M. Strozer, P. Gross, C.M. Aegerter, G. Maret, Observation of the critical regime nearAnderson localization of light. Phys. Rev. Lett. 96, 063904 (2006)

324. A. Suli, Motion indicators in the 2D standard map. PADEU 17, 47–62 (2006)325. A. Szell, B. Erdi, Z. Sandor, B. Steves, Chaotic and stable behaviour in the Caledonian

symmetric four-body problem. Mon. Not. R. Astron. Soc. 347, 380–388 (2004)326. M. Terraneo, M. Peyrard, G. Casati, Controlling the energy flow in nonlinear lattices: a model

for a thermal rectifier. Phys. Rev. Lett. 88, 094302 (2002)

252 References

327. U. Tirnakli, C. Beck, C. Tsallis, Central limit behavior of deterministic dynamical systems.Phys. Rev. E 75, 040106 (2007)

328. U. Tirnakli, C. Tsallis, C. Beck, Closer look at time averages of the logistic map at the edgeof chaos. Phys. Rev. E 79, 056209 (2009)

329. M. Toda, Theory of Nonlinear Lattices, (2nd edn.) (Springer, Berlin, 1989)330. S. Trillo, W. Torruellas (eds.), Spatial Solitons (Springer, Berlin, 2001)331. A. Trombettoni, A. Smerzi, Discrete solitons and breathers with dilute Bose-Einstein

condensates. Phys. Rev. Lett. 86, 2353–2356 (2001)332. C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World

(Springer, New York, 2009)333. C. Tsallis, U. Tirnakli, Nonadditive entropy and nonextensive statistical mechanics – Some

central concepts and recent applications. J. Phys. Conf. Ser. 201, 012001 (2010)334. G.P. Tsironis, An algebraic approach to discrete breather construction. J. Phys.

A-Math. Theor. 35, 951–957 (2002)335. S. Umarov, C. Tsallis, S. Steinberg, On a q-central limit theorem consistent with nonextensive

statistical mechanics . Milan J. Math. 76, 307–328 (2008)336. S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, Generalization of symmetric ˛-stable

Levy distributions for q > 1. J. Math. Phys. 51, 033502 (2010)337. A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev, Nonlinear oscillations of rarified plasma. Nucl.

Fusion 1, 82–100 (1961)338. H. Veksler, Y. Krivolapov, S. Fishman, Spreading for the generalized nonlinear Schrodinger

equation with disorder. Phys. Rev. E 80, 037201 (2009)339. H. Veksler, Y. Krivolapov, S. Fishman, Double-humped states in the nonlinear Schrodinger

equation with a random potential. Phys. Rev. E 81, 017201 (2010)340. N. Voglis, G. Contopoulos, Invariant spectra of orbits in dynamical systems. J. Phys.

A-Math. Gen. 27, 4899–4909 (1994)341. N. Voglis, G. Contopoulos, C. Efthymiopoulos, Method for distinguishing between ordered

and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372–377 (1998)342. G. Voyatzis, S. Ichtiaroglou, On the spectral analysis of trajectories in near-integrable

Hamiltonian systems. J. Phys. A-Math. Gen. 25, 5931–5943 (1992)343. J.-S. Wang, B. Li, Intriguing heat conduction of a chain with transverse motions. Phys. Rev.

Lett. 92, 074302 (2004)344. E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, 4th edn. (Cambridge University

Press, Cambridge, 1927)/(Cambridge Mathematical Library, Cambridge, 2002)345. D.S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, Localization of light in a disordered

medium. Nature 390, 671–673 (1997)346. S. Wiggins, Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990)347. S. Wiggins, Chaotic Transport in Dynamical Systems (Springer, New York, 1992)348. N. Yang, G. Zhang, B. Li, Carbon nanocone: a promising thermal rectifier. Appl. Phys. Lett.

93, 243111 (2008)349. H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268

(1990)350. H. Yoshida, Recent progress in the theory and application of symplectic integrators. Celest.

Mech. Dyn. Astr. 56, 27–43 (1993)351. K. Yoshimura, Modulational instability of zone boundary mode in nonlinear lattices: rigorous

results. Phys. Rev. E 70, 016611 (2004)352. N.J. Zabusky, M.D. Kruskal, Interaction of “solitons” in a collisionless plasma and the

recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)353. G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371,

461–580 (2002)354. Y. Zou, D. Pazo, M.C. Romano, M. Thiel, J. Kurths, Distinguishing quasiperiodic dynamics

from chaos in short-time series. Phys. Rev. E 76, 016210 (2007)355. Y. Zou, M. Thiel, M.C. Romano, J. Kurths, Characterization of stickiness by means of

recurrence. Chaos 17, 043101 (2007)

Index

Action-angle variables, 21, 107Additivity, 10Anderson localization, 165, 180, 181Area-preserving map, 212, 213Asymptotic stability, 3Autonomous dynamical system, 2

Bifurcation, 43Bifurcation theory, 6Bose Einstein condensation (BEC), 47, 61, 62Bushes of NNMs, 72, 82, 84Bushes of orbits, 70, 80

Canonical coordinates, 7Canonical transformation, 20, 22, 25Cantor set, 33Chaos, 2, 8, 30Chaotic motion, 99Chaotic orbit, 24, 34, 53, 57, 95–97, 104, 112,

114Compactness index, 184Complex instability, 43Conditional stability, 3Conservative dynamical system, 3, 58Constant of motion, 7Control methods, 176, 179Cyclic group, 71

Delocalization, 9Deviation vector, 53, 57–59, 92, 94, 103, 104,

106–108Diffusion, 9, 139, 157, 158, 160, 229Dihedral group, 71, 72

Discrete breather, 9, 135, 165, 166, 174Discreteness, 135, 166Discrete nonlinear Schrodinger (DNLS)

equation, 169Discrete symmetries, 12, 67Disorder, 9, 180, 183, 187Disordered discrete nonlinear Schrodinger

(DDNLS) equation, 183Dissipation, 36, 58Duffing oscillator, 30, 31, 38, 39, 61Dynamical system, 2

Edge of chaos, 9, 205, 213Elliptic fixed point, 16, 23, 34Energy, 7Energy surface, 8, 19Equipartition, 9, 67Escape energy, 30Extended primitive cells, 73Extensive quantity, 10, 56, 194Exterior algebra, 124Exterior product, 124

Feedback control, 176Fermi Pasta Ulam (FPU) system, 9, 46, 61, 65,

77, 117, 133, 136, 139, 197paradox, 134, 136, 138recurrence, 9, 52, 66, 67, 222

Fixed boundary conditions, 48Fixed point, 2, 3, 15, 34, 42Floquet theory, 45FPU. See Fermi Pasta Ulam (FPU) systemFrequency, 8Fundamental matrix, 42, 45

T. Bountis and H. Skokos, Complex Hamiltonian Dynamics,Springer Series in Synergetics, DOI 10.1007/978-3-642-27305-6,© Springer-Verlag Berlin Heidelberg 2012

253

254 Index

Generalized alignment index (GALI), 9, 59,102, 144, 155

Global stability, 3Group theory, 67

Hamiltonian function, 7Hamiltonian matrix, 42Hamiltonian system, 7Harmonic energies, 66Harmonic oscillator, 8, 13, 19Hartman-Grobman theorem, 18Heat conduction, 231Henon-Heiles system, 22, 27, 28, 61, 97, 111Heteroclinic orbit, 31, 33, 35Heteroclinic tangle, 33Hill’s equation, 50, 189Homoclinic orbit, 31–33, 35, 171, 172Homoclinic tangency, 218Homoclinic tangle, 33

In-Phase Mode (IPM), 48Instability, 6Integrability, 3Integral of motion, 3, 14, 21, 22, 30Integration by quadratures, 15, 20–22, 25Invariant manifold, 18Invariant torus, 22

Jacobian matrix, 6, 16, 93Jacobi elliptic functions, 16, 25

Kinetic theory, 236Klein-Gordon (KG) system, 167, 183Kolmogorov Arnol’d Moser (KAM) theorem,

x, 33Kolmogorov entropy, 8Kolmogorov-Sinai entropy, 55Korteweg-de Vries equation, 135

Lennard-Jones potential, 84, 88Linear stability, 16Liouville Arnol’d (LA) theorem, x, 18Liouville’s theorem, 33Localization, 9, 134, 139, 147, 149, 152, 165,

166Localization length, 182Local stability, 42Long range interactions, 193, 208Low-dimensional torus, 9, 122–124, 138

Lyapunov characteristic exponent (LCE), 5, 8,53, 98, 104

Lyapunov characteristic number, 5

Mathieu equation, 79Maximum Lyapunov exponent (MLE), 55Mel’nikov’s theory, 35, 36Melting transition, 208, 210Metal-insulator transition, 181, 182Metastability, 138Metastable states, 137, 213Method of Lyapunov functions, 7Microplasma, 208, 211Monodromy matrix, 45, 178Multibreather, 166, 170Multi-dimensional bushes, 67

Nonextensive statistical mechanics, 193, 212Non-integrable Hamiltonian, 28Nonlinear normal mode (NNM), 8, 48, 65Nonlinear oscillator, 9, 16, 22Normal coordinate, 76Normal mode, 8, 22, 63, 76

coordinates, 8eigenvectors, 181, 184, 186

Octahedral molecule, 84, 85One and a half degrees of freedom system, 31One-dimensional bushes, 67One-dimensional lattice, 64Order, 8Ordered motion, 11, 57Ordered orbit, 57, 95, 96, 100, 107, 113, 115Out-of-Phase Mode (OPM), 48

Painleve analysis, 24, 25, 27, 28Painleve property, 27, 35Parent symmetry group, 71, 72Participation number, 184Pendulum, 15Periodic boundary conditions, 48Periodic orbit, 7, 8, 24, 34, 35, 44Phase space, 2, 14, 16, 92Phonon band, 135, 166Poincare-Linstedt series, 136, 138, 140, 143,

145, 148, 152, 155, 164Poincare map, 32, 34, 35, 44Poincare surface of section (PSS), 29, 31, 35,

44, 48, 57, 99Poisson bracket, 25Poles, 26

Index 255

q-breather, 136–138, 149–151, 155, 156q-Gaussian distribution, 9, 11, 193, 195, 196,

204, 207, 208, 213, 216q-torus, 138–140, 146, 147, 149, 151–154, 156Quasilinear theory, 235Quasiperiodic orbit, 9, 22, 24, 29, 30, 33, 70Quasi-stationary state (QSS), 9, 192, 195, 212,

230

Resonance, 6Root modes, 75

Saddle fixed point, 16, 31, 32, 34, 35Secondary bush modes, 75Secular terms, 141Separatrix, 18, 31, 35Simple periodic orbit (SPO), 47Singularities, 24Smale horseshoe, 33Small divisors, 145, 147Smaller alignment index (SALI), 58, 94Solid state physics, 80Soliton, 135, 224, 233Specific energy, 137Spectrum of LCEs, 53Square molecule, 80, 82Stability, 3, 6, 8Stable and unstable manifolds, 32, 35, 43Stable fixed point, 16, 31Stable periodic orbit, 45Stickiness, 9, 213

Strong chaos, 8, 53, 192, 211Subdiffusive spreading, 183, 186, 187Symmetry group, 71, 72Symplectic integrators, 93Symplectic map, 9, 91, 93, 95, 96, 100, 103,

111, 120Symplectic matrix, 42, 46Symplectic structure, 41

Tangent map, 91, 93, 94Tangent map (TM) method, 93Tangent space, 25, 91, 92Thermal equilibrium, 10Thermodynamic limit, 55Thouless conductance, 182Topological equivalence, 18Transfer matrix method, 182Translational invariance, 64Tsallis entropy, 10, 194

Unstable fixed point, 16Unstable manifold, 18, 172Unstable periodic orbit, 45

Variational equations, 45, 53, 91, 92, 107Vibrational regime, 82

Weak chaos, 9, 51, 52, 192, 196, 205, 210Wedge product, 59, 102, 108, 124

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